Keywords: Problèmes Poisson and Stokes problems, Dirac measures, finite element methods, local error estimates, cilia, muco-ciliary transport

Mes premiers remerciements vont à mes directeurs de thèse. Merci Bertrand de m'avoir proposé ce sujet passionnant, et merci pour tes nombreuses et précieuses idées qui m'ont permis d'avancer chaque fois que j'en avais besoin. C'est toi aussi qui m'a dirigé vers Astrid et Sébastien. Je vous remercie chaleureusement tous les deux pour votre encadrement complémentaire qui a été très bénéfique pour moi. Vous avez toujours été très disponibles, sans compter les heures, et avez répondu à (toutes) mes questions et incertitudes. Je vous suis profondément reconnaissant pour votre indéfectible soutien tant au niveau enseignement et recherche qu'au niveau personnel. Je n'aurais pas pu espérer meilleurs directeurs de thèse, et ce à tous les niveaux.

J'adresse mes plus sincères remerciements à Franck Boyer et Emmanuel Maitre qui ont eu la gentillesse de rapporter cette thèse. Desidero ringraziare Angelo Iollo per avere accetato di fare parte della commissione, et je remercie aussi Philippe Poncet de me faire l'honneur de complèter ce jury.

Je remercie tous ceux avec qui j'ai eu l'occasion de collaborer durant ces quatres années, à commencer par les membres de l'équipe ANEDP du Laboratoire de Mathématiques d'Orsay. Je remercie en particulier Loïc Gouarin avec qui j'ai longuement travaillé pendant ma thèse et sans qui je n'aurais jamais pu me familiariser avec le code CAFES. Je le remercie aussi pour l'ensemble de ses interventions lors de mes nombreux "soucis informatiques", remerciements qu'il partagera avec Sylvain Faure que j'ai embêté au moins aussi souvent que lui. J'en profite pour remercier également Frédéric Lagoutière avec qui j'ai eu le plaisir de partager la majeure partie de l'enseignement pendant mon monitorat, et qui m'a peut-être plus appris sur le cours de "Mécanique et Modélisation" que nous en avons appris aux étudiants.

Je remercie l'ensemble de l'équipe du MAP5 qui m'a si bien accueilli lors de cette quatrième et dernière année de thèse. À cet égard, je remercie en particulier Annie Raoult et Clémence Misseboukpo qui ont largement oeuvré en ma faveur. J'y ai trouvé une équipe pédagogique fort sympathique et avec qui ce fut un plaisir d'enseigner.

La liste de mes collaborateurs ne serait pas complète si je ne remerciais pas Silvia Bertoluzza. Merci Silvia pour l'intérêt que tu as porté à notre travail et surtout le rôle clé que tu as joué dans la résolution de notre problème.

Je remercie Nathalie, Valérie, Catherine, Estelle, Florence et Marie-Hélène, qui ont rendu les tâches administratives beaucoup moins pénibles et chronophages.

Parce qu'un cadre de travail serein et agréable est plus que nécessaire pour préparer une thèse dans de bonnes conditions, je remercie l'ensemble de mes collègues doctorants et docteurs d'Orsay et du MAP5. Les moments les plus difficiles ont vite été effacés par la bonne ambiance et la cohésion des différents bureaux. Un merci spécial à mon petit frère de thèse Fabien avec qui j'aurais dû travailler mais que je laisse de toute façon entre de bonnes mains.

Merci à Ém

♥ ıl ♥ ıe pour notre "duo de choc" au séminaire des doctorants.

Un grand merci à Céline et Mélina qui diront sans doute qu'elles n'ont "rien fait du tout", mais dont les conseils avisés m'ont permis de si bien réussir mon audition de PRAG.

Je tiens à remercier l'ensemble de mes amis du basket et du tennis. Dès mon arrivée à Paris pour le basket, et un peu plus tard quand je me suis mis au tennis, j'ai trouvé comme une deuxième famille, si bien que mes longues soirées de rédaction où je n'ai pas pu vous voir cette dernière année, comme j'avais l'habitude de le faire avant, ont été d'autant plus difficiles pour moi. Malgré tout, votre soutien et votre compréhension auront été sans faille. Des remerciements particuliers pour Éric, Pascal, Paulo et Vincent dont le brunch du dimanche midi va me manquer. Je remercie aussi le groupe Anne-Laure, David, Massimo et les deux Nicolas pour les soirées plus folles les unes que les autres, et surtout Grégoire pour tout ce que tu m'as apporté. J'ajouterai Élodie et Justine, mon duo de rousses préféré, et Clémence avec qui je refaisais le monde régulièrement. Enfin, je remercie Élise et Morty pour ce qu'elles sont pour moi, et sans qui les vacances de Noël et d'été ne seraient pas pareilles. Bref, je remercie toute la sphère non mathématique qui m'entoure, qui ne comprend pas grand chose à ce que je fais ("Tu l'as démontré à qui le théorème ?") mais qui m'apporte l'équilibre dont j'ai besoin.

Je voudrais aussi remercier ma famille au grand complet. En particulier ma marraine, avec qui j'ai tant partagé, des soirées théâtre aux fêtes de Mont De quand j'étais censé lire et/ou rédiger des articles, et ma soeur, que je ne peux dissocier de mes deux nièces Jade et Lana, ou devrais-je dire mes "bouffées d'oxygène" à chaque fois que je redescends en vacances. Vos encouragements répétés ont eu beaucoup plus de poids que vous ne l'imaginez tout au long de mes (longues) études. Je remercie aussi mes cousins, Carole et Benjamin, dont les messages de soutien à chacune des étapes importantes m'ont toujours fait du bien (et je ne parle même pas des parties endiablées de tarot !). Enfin mes parents, vous m'avez toujours fait confiance, et je suis fier de pouvoir aujourd'hui vous raconter mon charabia.

Last but not least, mes derniers remerciements sont pour Christèle. Difficile de résumer en quelques lignes tout ce que tu m'as apporté chaque jour pendant ma thèse, merci d'être toi, tout simplement.

Modélisation et simulation du mouvement de structures fines dans un fluide visqueux : application au transport mucociliaire

Résumé

Une grande part des muqueuses à l'intérieur du corps humain sont recouvertes de cils qui, par leurs mouvements coordonnés, conduisent à une circulation de la couche de fluide nappant la muqueuse. Dans le cas de la paroi interne des bronches, ce processus permet l'évacuation des impuretés inspirées à l'extérieur de l'appareil respiratoire.

Dans cette thèse, nous nous intéressons aux effets du ou des cils sur le fluide, en nous plaçant à l'échelle du cil, et on considère pour cela les équations de Stokes incompressible. Due à la finesse du cil, une simulation directe demanderait un raffinement important du maillage au voisinage du cil, pour un maillage qui évoluerait à chaque pas de temps. Cette approche étant trop onéreuse en terme de coûts de calculs, nous avons considéré l'asymptotique d'un diamètre du cil tendant vers 0 et d'une vitesse qui tend vers l'infini : le cil est modélisé par un Dirac linéique de forces en terme source. Nous avons montré qu'il était possible de remplacer ce Dirac linéique par une somme de Dirac ponctuels distribués le long du cil. Ainsi, nous nous sommes ramenés, par linéarité, à étudier le problème de Stokes avec en terme source une force ponctuelle. Si les calculs sont ainsi simplifiés (et leurs coûts réduits), le problème final est lui plus singulier, ce qui motive une analyse numérique fine et l'élaboration d'une nouvelle méthode de résolution.

Nous avons d'abord étudié une version scalaire de ce problème : le problème de Poisson avec une masse de Dirac en second membre. La solution exacte étant singulière, la solution éléments finis est à définir avec précaution. La convergence de la méthode étant dégradée dans ce cas-là, par rapport à celle dans le cas régulier, nous nous sommes intéressés à des estimations locales. Nous avons démontré une convergence quasi-optimale en norme H s (s ě 1) sur un sous-domaine qui exclut la singularité. Des résultats analogues ont été obtenus dans le cas du problème de Stokes.

Pour palier les problèmes liés à une mauvais convergence sur l'ensemble du domaine, nous avons élaboré une méthode pour résoudre des problème elliptiques avec une masse de Dirac ou une force ponctuelle en terme source. Basée sur celle des éléments finis standard, elle s'appuie sur la connaissance explicite de la singularité de la solution exacte. Une fois données la position de chacun des cils et leur paramétrisation, notre méthode rend possible la simulation directe en 3d d'un très grand nombre de cils. Nous l'avons donc appliquée au cas du transport mucociliaire dans les poumons. Cet outil numérique nous donne accès à des informations que l'on ne peut avoir par l'expérience, et permet de simuler des cas pathologiques comme par exemple une distribution éparse des cils.

INTRODUCTION

Une grande part des muqueuses à l'intérieur du corps humain sont recouvertes de cils qui, par leurs mouvements coordonnés, conduisent à une circulation de la couche de fluide nappant la muqueuse. Dans le cas de la paroi interne des bronches, ce processus permet l'évacuation des impuretés inspirées à l'extérieur de l'appareil respiratoire : c'est la clairance mucociliaire (on parle aussi de transport mucociliaire). L'efficacité du transport est basée sur l'interaction entre les cils et le fluide environnant, ce qui rend complexe la modélisation mathématique de ce phénomène biologique. Mais mieux comprendre la rhéologie du système est capital pour l'étude de cas pathologiques, telle la mucoviscidose. La thèse porte sur la simulation directe en 3D du mouvement de cils dans un fluide visqueux. Par simulation directe, nous entendons prise en compte des mécanismes à l'échelle où ils se produisent, à savoir celle du cil. L'objectif est de décrire en détail le mouvement d'un ou plusieurs cils en interaction avec le fluide environnant, en se basant sur l'observation que le cil est une structure fine dont la fréquence de battement est très grande. Nous nous plaçons donc dans l'asymptotique où le rapport entre épaisseur et longueur du cil tend vers 0, mais où la résultante de la force appliquée par les cils sur le fluide reste constante. Cette approche conduit à l'étude de problèmes mathématiques singuliers, motivant une analyse numérique fine et l'élaboration d'une nouvelle méthode de résolution. L'enjeu est de développer un outil numérique pour la simulation du transport mucociliaire, capable en particulier de simuler un très grand nombre de cils battant dans un fluide complexe. Cet outil permettrait de mieux comprendre les mécanismes mis en jeu, et en jouant sur les différents paramètres, d'évaluer leur influence sur l'efficacité du processus de transport. Ce travail a bénéficié d'un financement public Investissement d'avenir, référence ANR-11-LABX-0056-LMH, LabEx LMH (Laboratoire de Mathématiques Hadamard).

Présentation du problème

Dans les poumons, la paroi des bronches est recouverte d'un film liquide visqueux dont le rôle est double : Introduction ' il capture les impuretés inhalées et les fait remonter le long des bronches, puis le long de la trachée, d'où elles basculent dans l'oesophage et sont éliminées dans l'estomac.

' il protège la paroi des bronches des agents pathogènes (particules de pollution, bactéries, etc.).

Ce film protecteur est composé de deux couches superposées. La première, contre la paroi des bronches, est la couche périciliaire (PCL, pour periciliary layer en anglais), essentiellement composée d'eau. Le mucus, la couche en contact avec l'air, est composé d'eau et de lipides, les mucines. Ces deux fluides forment une barrière d'une dizaine de micromètres d'épaisseur. Les impuretées piégées dans le mucus sont transportées hors de l'appareil respiratoire grâce à des cils baignant essentiellement dans la couche périciliaire, et dont le battement permet la propulsion du mucus vers le haut de la trachée (voir Figure 1 pour une photo des cils). L'ensemble de ce mécanisme est appelé clairance mucociliaire (ou transport mucociliaire).

Une des grandes difficultés dans l'étude du transport mucociliaire est le manque de données expérimentales. La structure dyadique de l'arbre bronchique rend impossible les mesures in vivo à partir de la sixième génération. Par ailleurs, ces mesures sont très peu précises car les techniques employées sont souvent trop invasives. La culture in vitro de cils n'est pas plus facile à cause de la dégradation du matériel biologique vivant après prélèvement [START_REF] Khelloufi | Physique de la dynamique mucociliaire. Dispositif d'étude de la migration cellulaire 3D. Application à l'asthme et à la BPCO[END_REF]. C'est pourquoi il est intéressant de développer un outil numérique donnant accès à des mesures de vitesses par exemple, que l'on ne peut que difficilement obtenir via l'expérience.

Une autre difficulté provient des différentes échelles de longueur que l'on peut considérer : moléculaire, ciliaire, bronchiale, etc. La Figure 1 illustre les différentes échelles mises en jeu.

Enfin, la complexité structurelle du transport mucociliaire rend son étude mathématique difficile. Beaucoup de paramètres sont à prendre en compte : ' le fluide nappant la paroi des bronches n'est pas homogène, il est composé de deux fluides de viscosités très différentes (le mucus est 50 fois plus visqueux que la couche périciliaire), et donc la dynamique de l'interface entre ces deux fluides peut faire l'objet d'une étude approfondie.

' les cils sont très fins, donc difficiles à prendre en compte numériquement, et la fréquence de battement est élevée.

' l'interaction entre l'air inspiré ou expiré et le mucus, sachant que les deux phénomènes ont des échelles de temps caractéristiques très différentes : les cils battent à une fréquence de l'ordre de 15Hz tandis que la fréquence de la respiration est de l'ordre de 0.25Hz.

' l'influence des agents pathogènes piégés dans le mucus qui peut aboutir à une modification de la rhéologie de l'écoulement bifluide.

Les travaux théoriques et numériques foisonnent sur le sujet et, malgré tout, la simulation du transport mucociliaire reste un enjeu important. La complexité du processus, dont Figure 1: Schéma multi-échelle du poumon humain, des premières générations de bronches (» 1 cm) aux cellules épithéliales (» 5 µm). Adapté à partir de [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF].

une liste non exhaustive des difficultés est donnée ci-dessus, rend le modèle numérique complet hors de portée des scientifiques à l'heure actuelle [START_REF] Causey | Mucus and the mare : how little we know[END_REF]. Dès lors, la compréhension du processus requiert une combinaison de méthodes qui incluent la modélisation mathématique, l'analyse des problèmes résultants et le calcul scientifique. Des modèles très différents mais complémentaires ont été proposés par les différentes auteurs, voir [START_REF] Smith | Modelling mucociliary clearance[END_REF] pour en avoir une synthèse détaillée.

L'objectif de la thèse est la simulation directe en 3D du mouvement d'un très grand nombre de cils dans un écoulement bifluide. La modélisation présentée dans la section suivante est guidée par la volonté de limiter les coûts de calculs, qui peuvent vite devenir exhorbitants, tout en essayant de faire le minimum d'hypothèses possible.

Introduction

Modèle fluide. Dans la littérature, les gens s'accordent sur le fait que la couche périciliaire est essentiellement de l'eau et a donc un comportement newtonien, même si cette affirmation a récemment été mise en cause par Boucher [START_REF] Boucher | Airway surface dehydration in cystic fibrosis : Pathogenesis and therapy[END_REF]. En revanche, pour le mucus, les modèles proposés peuvent être très différents : du modèle newtonien [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF], au modèle viscoélastique [START_REF] Mitran | Metachronal wave formation in a model of pulmonary cilia[END_REF][START_REF] Smith | A viscoelastic traction layer model of muco-ciliary transport[END_REF], et même parfois un fluide viscoplastique [START_REF] Craster | Surfactant transport on mucus films[END_REF]. Pour notre modèle, nous avons choisi de considérer un fluide newtonien, pour la couche périciliaire et pour le mucus. Cependant, nous tenons compte de la différence de viscosité entre les deux fluides, et modélisons l'ensemble par un modèle bifluide, dont l'interface entre les deux fluides est supposée plane et invariante (voir Figure 2). Cette hypothèse se vérifie expérimentalement [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF] et a été confirmée par les simulations [START_REF] Dillon | Fluid dynamic models of flagellar and ciliary beating[END_REF], même si cette hypothèse peut être relaxée. Re " ρLU µ , avec L et U respectivement la longueur et la vitesse caractéristiques de l'écoulement. Contrairement au nageur humain, pour lequel on a Re " 10 6 , la longueur caractéristique dans le cas du transport mucociliaire est le micromètre et la vitesse le micromètre par seconde, de sorte que le nombre de Reynolds est de l'ordre de 10 ´6, rendant ainsi les termes inertiels négligeables dans les équations de Navier-Stokes. On considère donc les équations de Stokes " ´µ△u `∇p " F e , divpuq " 0.

Ces équations étant complètement réversibles en temps, les cils bronchiques doivent produire un battement non réversible en temps sous peine d'avoir un effet moyen nul sur le fluide au cours d'une période.

Modèle pour le cil. Le but de cette thèse est de simuler en 3D et de manière directe toute une forêt de cils bronchiques, pour mesurer leur effet sur le mucus. La faisabilité des calculs, autrement dit éviter des coûts numériques exhorbitants, constitue notre ligne de conduite dans le choix du modèle pour le cil. Les deux modèles, dits de "cils discrets", les plus intuitifs sont :

' le domaine fluide exclut les cils, et on impose des conditions d'adhérence sur le bord de chaque cil u f luide " u cil . et Blake [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF]. C'est une décompositon en série de Fourier du battement de cil à partir d'images obtenues par Sanderson et Sleigh [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF]. Plus précisément, à chaque pas de temps t, le cil est représenté par la courbe paramétrée ξps, tq " L [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF]. b. Paramétrisation établie par Fulford et Blake [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF].

La Figure 4 illustre bien la correspondance entre les travaux expérimentaux [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF] et les travaux théoriques [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF] concernant la description du battement d'un cil bronchique. On peut déjà remarquer que le cil traverse l'interface mucus-couche périciliaire à l'aller, pendant la phase effective, mais reste complètement immergé pendant la phase de retour. Cette remarque est primordiale : le mouvement du cil n'est pas à moyenne nulle, ce qui est accentué par cette pénétration du cil dans le mucus, l'irréversibilité en temps étant nécessaire dans un fluide de Stokes pour permettre un transport du mucus efficace.

Paramétrisation d'une forêt de cils. L'article de Fulford et Blake [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF] indique aussi comment paramétrer toute une forêt de cils. Deux paramètres sont à prendre en compte : l'espacement ℓ 0 entre deux cils et la longueur d'onde λ de l'onde métachronale qui naît de la synchonisation des cils. Considérons une forêt de n x ˆny cils, où n x et n y sont respectivement les nombres de cils dans les direction x et y, le cil pi, jq est représenté par ξ i,j ps, tq " iℓ 0 e x `jℓ 0 e y `ξ ˆs, iℓ 0 f λ `t˙, Dans notre modèle, comme c'est toujours le cas dans la littérature, on supposera que cette onde métachronale se propage seulement dans la direction du transport de mucus (direction x), mais dans le sens opposé. Distribution de forces le long du cil. Pour définir la distribution de forces le long du cil, deux options s'offrent à nous : la première est la "resistive-force theory", développée par Gray et Hancock [START_REF] Gray | The propulsion of sea-urchin spermatozoa[END_REF] dans le cas de flagelles, qui suppose que la force hydrodynamique est proportionnelle à la vitesse locale du cil, mais cette théorie reste inconsistante dans le cas où les effets visqueux sont dominants [START_REF] Lighthill | Flagellar hydrodynamics[END_REF] ; la seconde, que nous avons choisie, est la "slender-body theory". Notre modèle est dû à Cox [START_REF] Cox | The motion of long slender bodies in a viscous fluid. Part 1. General theory[END_REF] qui a établi un développement asymptotique de la force lorsque le rapport L{r 0 tend vers l'infini (longueur du cil divisée par son rayon). Toujours en notant ξ la paramétrisation du cil, l'expression de la force en chaque point est approchée par fps, tq " 2πµ lnpL{r 0 q ˜2I 3 ´9 ξps, tq b 9 ξps, tq } 9 ξps, tq} 2 ¸ucil ps, tq, où 9

ξ est la dérivée de ξ par rapport à s et u cil est la dérivée par rapport à t. Cette expression a été établie en confrontant deux approches : la première consiste à considérer la vitesse tellement proche du cil qu'on distingue son rayon mais sa longueur L paraît tendre vers l'infini ; la seconde au contraire considère la vitesse loin du cil, sa longueur est donc finie mais son diamètre paraît nul tellement il est fin. Dans les deux cas, le rapport L{r 0 tend vers l'infini.

Conditions aux bords. Nous allons maintenant fixer les conditions aux bords. La boîte de calcul que l'on considère est un parallèlépipède rectangle dont le bord inférieur Introduction correspond à la paroi des bronches et la bord supérieur à l'interface entre le mucus et l'air. On suppose que la couche périciliaire adhère à la paroi des bronches, donc on y impose une vitesse nulle. Comme c'est souvent le cas dans les divers travaux menés sur le sujet, on suppose que la surface supérieure du mucus reste plane, ce qu'on modélise par une condition de non sortie du fluide et un glissement sans frottements (on néglige les effets de l'air sur le mucus). À l'échelle des cils, la fenêtre de calcul se focalise sur une longueur d'onde de l'onde métachronale, c'est-à-dire une centaine de cils. Les conditions aux bords les plus naturelles dans la direction du transport du mucus et celle qui lui est perpendiculaire sont donc des conditions bipériodiques, de façon à prendre en compte le battement des cils à l'extérieur de la boîte. Les conditions aux bords considérées sont rassemblées en Figure 6. Problème mathématique résultant. Ce paragraphe est la conclusion de l'ensemble de la modélisation présentée précédemment. Le modèle considéré, par linéarité des équations de Stokes, conduit au problème mathématique suivant :

$ ' & ' % ´µ△u `∇p " δ x 0 F on Ω, divpuq " 0 on Ω,
u " 0 on BΩ, (0.0.1) où x 0 est le point du domaine qui porte la force ponctuelle F. Les conditions aux bords décrites précédemment n'ont pas été prises en compte ici, mais elles le seront lors des calculs. En fait, les conditions aux bords choisies ne changent pas la régularité de la solution du problème résultant, et c'est pourquoi, dans un souci de simplicité et de clarté, l'analyse numérique sera étudiée en considérant des conditions aux bords de type Dirichlet homogènes. De même, si la viscosité variable peut poser des soucis pour la résolution numérique du problème, ce problème est indépendant du problème de la singularité du second membre. Ainsi s'affranchit-on de ces difficultés lors de l'étude théorique du problème de Stokes avec une terme source singulier

Analyse numérique de problèmes elliptiques singuliers

Le modèle présenté au paragraphe précédent a été choisi pour pouvoir simuler de manière directe toute une forêt de cils en 3D. Le modèle permet en effet de s'affranchir des conditions de raffinement de maillage puisqu'un maillage (structuré ou non) peut être envisagé indépendamment du nombre de cils. En contrepartie, un soin extrême doit être apporté à l'analyse numérique du problème résultant qui n'est pas usuel : le terme source du problème de Stokes est une force ponctuelle, terme source singulier en 2D et plus encore en dimension 3. Avant de s'intéresser au problème de Stokes, l'analyse numérique est menée dans le cas du problème de Poisson avec une masse de Dirac en second membre : ce problème est la version scalaire du problème de Stokes et, tout en étant plus simple, plusieurs résultats obtenus dans ce cas sont adaptables au problème de Stokes singulier.

Analyse numérique du problème de Poisson singulier. Le problème que l'on considère est le suivant " ´△u " δ x 0 dans Ω u " 0 sur BΩ, où Ω est un domaine régulier et x 0 est un point de ce domaine. La solution de ce problème n'étant pas dans H 2 pΩq, les résultats classiques de convergence des méthodes éléments finis ne sont plus garantis, à savoir une convergence à l'ordre 1 en norme H 1 pΩq et à l'ordre 2 en norme L 2 pΩq. En fait la solution n'est même pas dans H 1 pΩq, ce qui fait que les méthodes ne sont a priori pas définies. Si l'on considère des éléments finis continus (comme c'est très souvent le cas), la solution numérique a bien un sens et on peut alors s'intéresser à la convergence de cette solution numérique vers la solution exacte. Ce problème a largement été étudié dans la littérature. Parmi les résultats principaux on compte celui de Scott [START_REF] Scott | Finite element convergence for singular data[END_REF], qui a montré une convergence en norme L 2 pΩq en h d{2 où h est la taille caractéristique du maillage et d la dimension. Au moyen d'un raffinement du maillage autour de la singularité, Apel et ses co-auteurs [2] ont recouvré une convergence L 2 à l'ordre 2 en 2D, mais avec un maillage équivalent à du h 2 près de la singularité : ce résultat rejoint celui de Scott mais apporte l'information qu'un raffinement de maillage local seulement suffit.

De notre côté, nous nous sommes intéressés au comportement de la solution numérique "loin" du Dirac, c'est-à-dire que nous avons étudié la convergence de la méthode sur un sous-ensemble Ω 0 qui exclut la singularité. Parallèlement à des travaux de Köppl et Wohlmuth [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF], qui ont montré une convergence optimale en norme L 2 pΩ 0 q, nous avons prouvé une convergence quasi-optimale en norme H s pΩ 0 q, s ě 1. Plus précisément, le résultat que nous avons prouvé en 2D est le suivant :

Theorem. Soient Ω 0 ĂĂ Ω 1 ĂĂ Ω tels que x 0 R Ω 1 et 1 ď s ď k. On note u la solution du problème de Poisson avec un Dirac en second membre, et u h sa projection de Galerkin sur l'espace éléments finis V k h . Alors, il existe h 1 tel que pour tout 0 ă h ă h 1 , }u ´uh } 1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k a | ln h|, où k est l'ordre d'approximation de l'espace V k h . De plus, pour s ě 2, si on suppose en plus que les éléments finis sont H k -conformes, on a }u ´uh } s,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k`1´s a | ln h|.

Introduction

Ce résultat est vrai dès que l'espace éléments finis V k h vérifie quelques hypothèses très classiques et vérifiées par une grande classe d'espaces éléments finis, incluant tous les éléments finis définis sur des maillages quasi-uniformes. En particulier, les éléments finis de Lagrange et de Hermite, ou encore les ondelettes vérifient ces hypothèses. Elles sont écrites plus en détail dans le Chapitre 2.

La preuve de ce théorème s'appuie fortement sur le Théorème de Nitsche et Schatz [START_REF] Nitsche | Interior estimates for Ritz-Galerkin methods[END_REF], qui donne une estimation de l'erreur locale optimale (comme dans le cas régulier) moyennant l'erreur dans une norme aussi faible que l'on veut. Pour estimer cette erreur en norme faible, nous avons démontré une version faible du Lemme d'Aubin Nitsche. On se ramène alors au contrôle de l'erreur en norme W 1,q pΩq, avec q ă 2, que l'on obtient grâce à une égalité inf-sup discrète.

En 3D, le résultat obtenu n'est plus quasi-optimal puisque la convergence que l'on obtient n'est plus en h k a | ln h| mais en h k´1 3 a | ln h| 2 . En fait, les arguments donnés en 2D fonctionnent car la solution est dans W 1,2´τ pΩq, pour tout τ ą 0, c'est-à-dire à la limite d'être dans H 1 pΩq. En 3D, la solution est dans W 1,3{2´τ pΩq, elle est bien plus singulière, et cela suffit à mettre en défaut la preuve présentée en 2D.

Dans le cas particulier des éléments finis de Lagrange P 1 , nous avons montré un résultat légèrement plus fort : }u ´uh } 1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh.

Ce résultat s'obtient encore grâce au Théorème de Nitsche et Schatz, mais avec d'autres arguments que ceux présentés dans le cas précédents : la clé réside en le fait que les fonctions test sont harmoniques sur chaque maille. Malheureusement, cette preuve ne s'étend ni au cas des éléments finis P k , k ě 2, ni au cas de la dimension 3, ni au problème de Stokes.

Analyse du problème de Stokes singulier. Comme dit dans le préambule de cette partie, certains résultats obtenus dans le cas du problème de Poisson singulier peuvent être montrés dans le cas du problème de Stokes avec une force ponctuelle en terme source. Cette fois-ci encore, la solution n'est pas suffisamment régulière pour espérer une convergence optimale des méthodes éléments finis sur l'ensemble du domaine. À l'image de l'étude menée dans le cas du problème de Poisson, nous nous sommes intéressés aux erreurs locales des méthodes éléments finis classiques, et plus particulièrement aux éléments P k {P k´1 , pour k ě 2, et P 1 b{P 1 pour le cas k " 1. Le résultat que nous avons montré toujours en 2D est le suivant :

Theorem. Soient Ω 0 ĂĂ Ω 1 ĂĂ Ω tels que x 0 R Ω 1 , k ě 1, 1 ď q ă 2, considérons pu, pq P W 1,q 0 pΩq ˆLq 0 pΩq la solution du problème de Stokes avec une force ponctuelle en terme source et pu h , p h q sa projection de Galerkin sur l'espace éléments finis V k h ˆW k h telle que ş Ω p h " 0 et ż Ω ∇pu ´uh q :: ∇η ´żΩ pp ´ph qdivpηq " 0 for all η P V k h , ż Ω divpu ´uh qξ " 0 for all ξ P W k h .

Une méthode numérique pour la résolution de problèmes elliptiques singuliers 27

Sous l'hypothèse pu, pq P H k`1 pΩ 1 q 2 ˆHk pΩ 1 q, il existe h 1 tel que pour tout 0 ă h ď h 1 , }u ´uh } 1,Ω 0 `}p ´ph } 0,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k a | ln h|.

La preuve est présentée dans le Chapitre 3. Elle s'appuie sur le Théorème d'Arnold et Liu [4], qui est un résultat équivalent à celui de Nitsche et Schatz, mais pour le problème de Stokes. Bien que l'ajout de la pression complique les choses, le squelette de la preuve est sensiblement le même que dans le cas du problème de Poisson. Quelques généralisations de ce résultat sont données à la suite de la preuve. Cependant, comme dans le cas du problème de Poisson, le second membre étant plus singulier encore, et les estimations obtenues en 3D ne sont plus quasi-optimales.

Une méthode numérique pour la résolution de problèmes elliptiques singuliers

Nous avons prouvé que la convergence des méthodes éléments finis classiques était optimale (ou quasi-optimale) sur un sous-domaine qui exclut la singularité. Mais nous savons aussi que sur l'ensemble du domaine la convergence est mauvaise. Pour pallier ces problèmes de convergence, nous proposons une nouvelle méthode numérique, dite de "soustraction", qui permet de retrouver un ordre de convergence optimal sur l'ensemble du domaine, sans raffinement. Cette méthode est basée sur la constat que, dans le cas de la masse de Dirac ou de la force ponctuelle, on connaît explicitement la singularité. Il suffit alors d'extraire la singularité de la solution et de se ramener à un problème auxiliaire régulier, dont l'analyse numérique conduit à une convergence optimale, quel que soit l'ordre d'approximation des éléments finis considérés.

Par exemple, dans le cas du problème de Stokes, une solution fondamentale est donnée par la Stokeslet pu δ , p δ q :

" ´µ△u δ `∇p δ " δ x 0 F dans R d , divpu δ q " 0 dans R d ,
où d est la dimension de l'espace, et F la force ponctuelle localisée au point x 0 . Pour extraire la singularité de la Stokeslet, concentrée au point x 0 , nous allons la multiplier par une fonction plateau χ régulière qui vaut 1 sur un voisinage de x 0 . Plus précisément, soient 0 ă a ă b ă distpx 0 , BΩq :

' χ est supposée régulière, χ P H 2`k pR d q. ' χ vaut 1 "près" de x 0 , χ | Bpx 0 , aq " 1.
' χ vaut 0 "loin" de x 0 , χ | Bpx 0 , bq c " 0.

Introduction

On pose alors u 0 " χu δ et p 0 " χp δ , et on calcule ´µ△u 0 `∇p 0 et divpu 0 q :

' sur Bpx 0 , aq, χ " 1 donc ´µ△u 0 `∇p 0 " ´µ△u δ `∇p δ " δ x 0 F et divpu 0 q " divpu δ q " 0.

' sur Bpx 0 , bq c , χ " 0 donc ´µ△u 0 `∇p 0 " 0 et divpu 0 q " 0.

En fait, pu 0 , p 0 q est solution de

$ ' & ' % ´µ△u 0 `∇p 0 " δ x 0 F `g dans Ω, divpu 0 q " h dans Ω, u 0 " 0 sur BΩ,
où les termes sources réguliers g et h sont connus explicitement et à support dans la couronne centrée en x 0 , de rayon intérieur a et de rayon extérieur b (voir Figure 7). Ainsi, en notant pv, qq la solution exacte du problème $ ' & ' % ´µ△v `∇q " ´g dans Ω, divpvq " ´h dans Ω, v " 0 sur BΩ, et v h la solution numérique associée, on obtient une solution numérique u h du problème initial en posant u h " u 0 `vh . La solution approchée u h ainsi définie converge vers la solution exacte u à l'ordre optimal, quel que soit l'ordre d'approximation des éléments finis utilisés, car l'erreur u´u h est égale à v ´vh , l'erreur obtenue en résolvant numériquement le problème auxiliaire régulier.

La méthode de soustraction a été présentée dans le cas du problème de Stokes mais peut être généralisée à beaucoup de problèmes elliptiques, dès lors qu'on en connaît une solution fondamentale (exemple : le problème de Poisson). Elle se généralise aussi à d'autres cas de conditions aux bords plus complexes que des conditions de Dirichlet homogènes, comme celles que nous avons fixées pour notre modèle de transport mucociliaire (voir Figure 6). Enfin, la méthode a été décrite dans le cas d'un fluide de viscosité constante, mais peut être facilement adaptée au cas d'un écoulement bifluide.

Simulation directe en 3D d'une forêt de cils

La méthode de soustraction présentée précédemment permet de simuler en 3D et de manière directe toute une forêt de cils : par linéarité du problème de Stokes, considérer une seule force ponctuelle ou un très grand nombre revient au même dans le sens où on ne résout qu'un seul problème numérique à chaque itération en temps. Pour les calculs, nous utilisons le code CAFES, écrit en C/C++ et développé par Benoît Fabrèges pendant sa Simulation directe en 3D d'une forêt de cils 29 thèse en collaboration avec Loïc Gouarin et Bertrand Maury. Le code est notamment utilisé pour sa partie solveur de Stokes parallèle et notre contribution a été l'implémentation de la méthode de soustraction décrite ci-dessus, de la construction du second membre à la correction de la solution numérique obtenue en résolvant le problème auxiliaire régulier.

Nous avons tout d'abord testé la méthode sur le cas modéré d'un petit paquet de cils : 15 dans la direction x et 15 dans la direction y (voir Figure 8). Notons quand même que ce "petit" cas test représente déjà beaucoup plus de cils que ce qu'ont pu considérer certains travaux [START_REF] Dillon | Fluid dynamic models of flagellar and ciliary beating[END_REF][START_REF] Lee | Muco-ciliary transport : effect of mucus viscosity, cilia beat frequency and cilia density[END_REF]. Pour ce premier test, on considère un fluide de viscosité constante : l'idée est d'observer la différence d'efficacité du transport pendant les phases effective et de retour des cils (voir Figure 8). Après avoir constaté l'efficacité de la méthode sur quelques centaines de cils, nous Introduction avons testé sa robustesse sur des milliers de cils. Pour le calcul suivant, on se place dans des conditions réelles : simulation directe en 3D d'une forêt de cils complète, battant dans un écoulement bifluide et avec les conditions aux bords réalistes. La Figure 9 illustre ce calcul qui sera en fait le cas témoin auquel les calculs suivants seront comparés. Deux remarques immédiates émergent de ce calcul : le mucus se déplace à vitesse constante tel un "bloc" glissant sur la couche périciliaire, et la vitesse dans la couche périciliaire croît linéairement, comme déjà observé dans divers travaux [START_REF] Matsui | Coordinated clearance of periciliary liquid and mucus from airway surfaces[END_REF][START_REF] Smith | Modelling mucociliary clearance[END_REF].

L'outil numérique que nous avons développé permet donc non seulement la simulation directe en 3D d'un très grand nombre de cils, mais aussi la simulation de certaines pathologies liées au processus de clairance mucociliaire. Nous avons pu, entre autres, simuler une hauteur de la couche périciliaire anormalement élevée (cas où les cils n'atteignent jamais le mucus), ou au contraire anormalement basse (cas où les cils atteignent le mucus aussi dans la phase de retour). L'influence de la densité de cils a aussi été étudiée, de façon à mesurer les effets de seuil de densité sur l'efficacité du transport.

Plan de la thèse

La thèse porte sur la modélisation et la simulation directe en 3D d'une forêt de cils. Elle est divisée en 4 chapitres, un chapitre de présentation et modélisation, 2 chapitres d'analyse numérique, et un dernier axé calcul scientifique.

Nous commençons le premier chapitre par une présentation d'ensemble du phénomène qui est étudié, à savoir la clairance mucociliaire. Les principales caractéristiques biologiques et mécaniques des cils, du mucus et de la couche périciliaire y sont décrits. Après un état de l'art des différentes études menées sur le sujet, nous y décrivons notre modèle pour le fluide et les cils. La fin du chapitre est consacrée à l'étude théorique de deux problèmes elliptiques singuliers : le problème de Stokes avec une force ponctuelle en terme source, problème résultant de la modélisation choisie, et le problème de Poisson avec une masse de Dirac en second membre, une version scalaire du problème de Stokes singulier.

Le deuxième chapitre est dédié à l'analyse numérique de la résolution du problème de Poisson avec une masse de Dirac en second membre par des méthodes éléments finis classiques. Plus précisément, on montre pour des éléments finis très généraux que la solution numérique converge à l'ordre quasi-optimal sur tout sous-domaine qui exclut la singularité. En d'autres termes, la convergence de la méthode est altérée par la singularité du second membre, mais l'erreur se concentre autour de la singularité, pour une bonne convergence "loin" du Dirac.

Des résultats analogues de convergence sont montrés dans le troisième chapitre dans le cas du problème de Stokes avec une force ponctuelle en terme source. Cette fois-ci, l'étude est menée dans le cas des éléments P k {P k´1 , pour k ě 2, et P 1 b{P 1 pour le cas k " 1, puis généralisée ensuite.

Le quatrième et dernier chapitre est divisé en deux grandes sections. La première présente une nouvelle méthode numérique, dite de soustraction, qui permet de recouvrir une convergence optimale sur l'ensemble du domaine quel que soit l'ordre d'approximation Plan de la thèse 31 des éléments. Cette méthode est ensuite généralisée à des cas plus compliqués comme un modèle bifluide, des conditions aux bords non homogènes, etc. La seconde section met en application cette méthode de soustraction et présente les résultats numériques obtenus dans le cas du transport mucociliaire. L'influence de plusieurs facteurs importants est testée comme l'importance de l'écoulement bifluide, la densité de cils ou la hauteur du mucus.

CHAPTER 1

FROM THE MUCOCILIARY TRANSPORT

TO THE STOKES PROBLEM The respiratory system is the largest organ of human body. The volume that it occupies (from 5 to 7 litres) is essentially air. Half a litre is blood and less than another half forms the various tissues. It is composed of:

• the structure areas, among them the rib cage, the diaphragm, intercostal muscles, pleural area...

• the bronchial tree, with the trachea, the bronchi and the bronchioli.

• the respiratory areas formed by the alveoli, where the gaseous exchanges with the blood occur.

Through the breathing process, its role is to supply the body with oxygen (distributed by the blood) and to evacuate the carbon dioxide it products. To help in this process, a maximal air-blood exchange area is necessary: the exchange area is the boundary of a huge collection of small balls (around 300 million units), called alveoli, the diameter of which is about a quarter millimeter. That makes an exchange area (membrane of those alveoli) of about 100 m 2 . The alveoli are connected to the outside world through the respiratory tract, which has a very particular spatial structure of tree (see Figure 1.1).

Figure 1.1: Bronchial tree mouding, created by E. R. Weibel [START_REF] Weibel | Morphometry of the human lung[END_REF]. Zoom on the little bronchi.

Indeed, the respiratory tract is an assembly of interconnected pipes following a very dyadic-tree structure. The overall tree can be described as follows: the trachea (with a diameter around 2 cm) divides into two sub-branches, which divide further onto smaller branches, and so on, up to around 23 levels of bifurcations. The word "around" is important: the aim is to occupy the whole available space in the rib cage, and some branches stop before the 23rd generation whereas others divide after [START_REF] Weibel | Design of peripheral airways for efficient gas exchange[END_REF]. The first generations are purely conductive, exchanges do not take place before generation 16 or 17. Beyond that point, all branches are lined up with alveoli, up to the last generation. With 300 million alveoli, the exchange area is around 100 m 2 in human adults.

As said previously, the exchange area must be sufficiently large to allow the required quantity of oxygen to be transferred by passive diffusion. Therefore preserving the lung from the inhaled impurities is necessary: they could obstruct the bronchi and limit the exchange area. The breathed products which could settle in the bronchopulmonary system are eliminated thanks to three principal mecanisms:

• the cough.

• the mucociliary transport (or mucociliary escalator).

• the alveolar clearance.

Although it is not the aim of this thesis to discuss the mecanism and results of coughing in any detail, the relationship of cough to mucociliary transport deserves a mention. During cough, the respiratory muscles of the rib cage and diaphragm contract to expel air violently from the lungs, and this expulsion is accompagnied by a narrowing of the larger airways and an oscillation of the tissues and linings of the airway wall [START_REF] Leith | Cough[END_REF]. Gases may be expelled from the mouth at 10 L.s ´1 or more, implying gas velocities well in excess of 10 m.s ´1, even in narrow airways, and estimated speeds in regions of airway compression range as high as 250 m.s ´1, which is three quarters of the speed of sound.

In a few words, the alveolar clearance is the process during which particles are removed from the alveoli. This clearance can take many forms: phagocytosis by macrophages, elimination of the dust-loaded macrophages towards the ciliated airways, detention in the interstitium and transport to the lymphatic nodes [START_REF] Lauweryns | Alveolar clearance and the role of the pulmonary lymphatics[END_REF].

In this thesis, we focus on the mucociliary transport and its mecanisms.

The mucociliary transport

The lining of the bronchi is napped with a mucus film whose role is to catch the inhaled impurities (dust, pollution particles, bacteria, fungus...). The mucus traps the pathogens, reduces microbial access to the epithelial cells, cells which compose the lining of the bronchi, and prevent them from accumulating in the bronchi and obstructing the lung. The mucus and the trapped impurities are transported outside the lung by means of the beat of submerged cilia, until the junction with the oesophagus to be eliminated in the stomach.

More precisely, at the scale of the cilium, on the lining of the bronchi (see Figure 1.2) the fluid is biphasic [START_REF] Lucas | Principles underlying ciliary activity in the respiratory tract : II. A comparison of nasal clearance in man, monkey and other mammals[END_REF]. The lower layer is the periciliary layer (commonly called PCL). It contains the cilia. The upper layer is the mucus. The lining epithelium of the respiratory tract is thicker in its upper parts, where it is underlain by a thick submucosal zone [START_REF] Schneeburger | The integrity of the air-blood barrier[END_REF]. The epithelia of ciliated regions are known to be complex, at least in part: in addition to the easily distinguishable ciliated cells, there are several types of epithelial cells, as described by Jeffrey and Reid [START_REF] Jeffrey | The respiratory mucous membrane[END_REF], among them the goblet cells responsible for secreting Modified from [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF].

the proteins which compose the mucus. In the large airways, the apices of ciliated cells may form a more or less continuous cover at the surface of the epithelium. For instance, the tracheal epithelium may have only 1 goblet cell for every 5 ciliated cells. But, in the smaller airways, the proportion increases and there may be numerous intermediate cells.

Mucociliary transport depends upon a successful relationship between these three components: the cilia, the mucus and the periciliary fluid. The characteristics of transport may be altered by changes in any of these three components, and such changes may be used to regulate transport or changes resulting from disease may interfere with transport. Let us introduce some detailed aspects of the mucociliary transport which is the focus of this work from the mathematical point of view.

The mucus

There are two different types of biological fluids: the ones which flow in a closed circuit, like for instance the blood and the lymph, and the ones which flow in an open circuit, like mucus, saliva, sperm... Although the functions of each of these fluids are very different, they are all essentially composed of water and some other specific components: proteins, cells, nutriments... Respiratory airway mucus is a complex mixture of several proteins (glycoproteins, proteoglycans, lactoferrin, lysozyme...), lipids and salt [START_REF] Coles | Airway mucus : composition and regulation of its secretion by neuropeptides in vitro[END_REF][START_REF] Lopez-Vidriero | Airway secretion : source, biochemical and rheologicol properties[END_REF]. This mixture imparts protective properties to the lung: some proteins enhances the capture of bacteria and reduces microbial access to the epithelia; some remove iron required by many bacteria while others kill these bacteria [START_REF] Clamp | Some non-mucin components of mucus and their possible biological roles[END_REF]. Microorganisms are also trapped by the various lipids attached to some proteins, and the more diverse the lipids, the wider their binding properties. Mucus macromolecules absorb many kinds of foreign molecules avidly, and sulphated groups interact with viruses.

The glycoproteins appear to be the most important component conferring viscoelastic properties to the mucus. The rheological behavior of the mucus is important [START_REF] Litt | Mucus rheology : Relevance to mucociliary clearance[END_REF][START_REF] Litt | Flow behavior of mucus[END_REF][START_REF] Lopez-Vidriero | Airway secretion : source, biochemical and rheologicol properties[END_REF]. When cilia beat in water, the rate of flow increases linearly with the force exerted by the cilia because the viscosity remains constant [START_REF] Sleigh | Movement and coordination of tracheal cilia and the relation of these to mucus transport[END_REF]; such fluids show Newtonian behavior. Some other fluids, including mucus, are non-Newtonian and behave as shear-thinning fluids in which the viscosity, although much greater than water, decreases as the applied force is increased, so the more forcefully the cilia beat, the more easily the mucus moves. But mucus has also elastic properties. When it is stretched and released, it may return to near its original shape using energy stored in it by the original deformation, although some energy is also dissipated in overcoming the viscous forces. The time taken to use up the stored energy against viscous resistance is a measure of the relaxation time, which relates the moduli of elasticity and viscosity. Experiments suggest that the relaxation time for mucus is quite long (perhaps 30 s [START_REF] Gilboa | In situ rheological characterization of epithelial mucus[END_REF]). Thus, when cilia, which beat at rates of 10 to 20 Hz, exert propulsive forces on the mucus, stretching forces are applied to the mucus at much shorter time intervals than the relaxation time. The mucus is therefore seen by the cilia as an elastic structure able to accept efficient energy transfer from the cilia and relaxing very little between successive beats. It has been observed that the mucus forms a blanket in the larger airways, but is more broken in smaller airways [START_REF] Hilding | Ciliary streaming in the lower respiratory tract[END_REF][START_REF] Sturgess | The mucus lining of major bronchi in the rabbit lung[END_REF]. In the smaller bronchioli, the mucus is present in the form of droplets [START_REF] Iravani | Mucus transport in the tracheobronchial tree of normal and bronchitic rats[END_REF]. These droplets are more numerous in the larger bronchioli and more likely in the larger airways to aggregate into flakes. In the bronchi and the trachea, the mucus seems to be a more continuous sheet. It appears possible that the mucus is normally only secreted in response to stimulation, and that the small droplets, flakes or plaques are bearing away trapped particules, perhaps bacteria or macrophages. Certainly one might expect a larger number of particules to be filtered out by inertial impact against the epithelium in the larger airways, and larger mucous plaques would be required to support them on the ciliary escalator. Let us note that flakes and plaques of mucus are more abundant in many infections and diseases of human respiratory tracts.

Finally, the mucus is the upper layer of the fluid napping the lining of the bronchi. It traps the inhaled impurities and transports them outside the lung. The mucus is a viscoelastic fluid but the relaxation time is long, so that we can model the mucus as a Newtonian viscous fluid.

The periciliary layer

The cilia of the respiratory tract beat in a low viscosity periciliary layer whose depth is a little less than the extended ciliary length when mucus is present. Although it is believed that the cilia achieve little net transport of periciliary fluid beneath the mucus, there is no quantitative data. Mathematical modeling suggests that at least the lower part of the periciliary layer shows minimal net flow. Clearly, the layer of fluid immediately below the mucus must flow at the same speed as the mucus, but we do not know anything about the relative depths of these two regions of the periciliary layer. Recently, it has been suggested that a further role of the PCL is to prevent the adherence of epithelial cells with the overlying mucus layer [START_REF] Knowles | Mucus clearance as a primary innate defense mechanism for mammalian airways[END_REF].

The activity of cilia may play a part in maintaining and controlling the depth of the periciliary layer. If the periciliary fluid layer becomes too deep, the cilia will be detached from the mucus during their effective strokes (see below), leading to lower transport rates until excess fluid is removed by cilia and they reach the mucus once again. In dry air, the loss of fluid by evaporation may reduce the depth of periciliary fluid and damage the cilia.

The cilia

The ciliary component is the best understood of the three components (mucus, periciliary layer and cilia). The first comprehensive account of cilia seems to be due to Sharpey in 1835 [START_REF]Cyclopaedia of anatomy and physiology[END_REF], who not only gave detailed descriptions of the actions of cilia in a wide variety of animals but also reported the discovery of ciliary motion in the respiratory systems of mammals. Since then a long period of active research, notably by Lucas who studied ciliary function in the upper respiratory tract [START_REF] Lucas | Principles underlying ciliary activity in the respiratory tract : II. A comparison of nasal clearance in man, monkey and other mammals[END_REF]. As well as providing detailed confirmation of the role of cilia in mucociliary transport, this work led to the important conclusion that mucus is propelled by the tips of the cilia which themselves move in a low viscosity layer beneath the mucus. [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF].

Length of cilium

Cilia fixed on the epithelial cells in the bronchi are thin structures whose length L is about 6 µm in the larger airways (reduced to 5 µm or a little less in the smaller bronchioli) and radius r 0 is 0.1 µm. They beat at an average frequency f of 15 Hz (see Table 1.1 for a summary of these data). We are talking about an "average frequency" because ciliary beat frequency depends on several parameters, among them the temperature and the humidity of the environment. Fluctuations in secretion of mucus can also stimulate the activity of the cilia [START_REF] Puchelle | Mucociliary frequency of frog palate epithelium[END_REF][START_REF] Spungin | Stimulation of mucus secretion, ciliary activity, and transport in frog palate epithelium[END_REF], secretion which can be increased or decreased by some drugs [START_REF] Nadel | New approaches to regulation of fluid secretion in airways[END_REF]. As it will be explained below, the cilia do not beat alone but take part of a metachronal wave which propagates to the surface of a carpet of 0.3 µm-spaced cilia with a wavelength of 30 µm.

The complete modelling of the polymerization of a cilium is still an open problem at this time. A sectional view of a cilium (see Figure 1.4) shows a normal 9 + 2 axonemal structure of fibrils: a doublet of fibrils (called "axoneme") is encircled with nine other doublets of fibrils. Moreover, they are unusual in the possession of a crown of 3 to 7 short "claws" 25 to 35 nm long (see Figure 1.3), projecting from a dense cap at their tips [START_REF] Foliguet | Apical structure of human respiratory cilia[END_REF]. The ciliary basal body is of a common type with a basal foot, short striated rootlets, and attached cytoplasmic microtubules, which together provide anchorage. This structure is characteristic of the cilia. Bending movements are produced as the outer nine microtubules doublets of the ciliary axoneme actively slide against one another when propelled by molecular bridges of dynein that project from one doublet towards the next. Dynein is a ATPase protein that uses energy from ATP in performing cyclical shape changes that produce the active sliding movements. The machinery of motility is spread along the whole length of the cilium, and different patterns of sliding along the length of the nine doublets are responsible for the differences in shape of the cilium during its beat. Radial connections between axonemal fibrils are assumed to resist the sliding and contribute to the formation of bends. A detailed description of the mechanism of motility has been given by Gibbons [START_REF] Gibbons | Cilia and flagella of eukaryotes[END_REF].

The beat of a cilium can be broken down into two parts: a recovery stroke and an effective stroke. These two phases does not counterbalance one with the other. This point is very important. Indeed, mucus and periciliary layer are viscous fluids at the scale of a cilium, and these fluids have reversibility property: a reversible movement of the cilium would not permit the mucus to be transported (see Section 1.3.1.2). Between two cycles, the cilium is at rest. At the start of the beat cycle, the cilium is in the rest position r (see Figure 1.5). In the recovery stroke the cilium unrolls clockwise (in the top view) from its rest position to the starting position for the effective strokes, remaining close to the cell surface. In the effective strokes, the cilium is fully extended and bends over to reach the rest position. Both side and top views show the beat is 3-dimensional, even if the effective stroke is nearly planar and in a plane perpendicular to the cell surface, during which the cilium swings through an arc of about 110 degrees.

The different motion of the cilium between the effective and recovery strokes can be well understood if we consider the motion of a needlelike body in a viscous liquid. The force acting on a needlelike body moving perpendicular to the axis of symmetry is almost twice the force acting on the same needlelike body moving in the axial direction. Thus, it would be highly desirable for the cilium to have a "perpendicular mode" of motion in its effective stroke (thus generating a greater force) as against a "tangential mode" of motion during the recovery strokes. In addition, the force is linearly dependent on the velocity; thus, a larger velocity implies a larger force, which is clearly obvious in the ciliary beat pattern with the fast effective strokes. Indeed, according to Sanderson and Dirksen [START_REF] Sanderson | A versatile and quantitative computer-assisted photoelectronic technique used for the analysis of ciliary beat cycles[END_REF], the effective stroke is two or three times faster than the recovery stroke and the rest phase. When mucus is present, floating on top of the periciliary layer, it comes within reach of the ciliary tips during the effective stroke of the ciliary beat cycle, but the cilia will move beneath the mucus for the rest of the cycle. At the start of the effective stroke, cilia are often still slightly bent forward and may therefore enter the overlying mucus the tip first rather than being bent backwards and presenting the side of the cilium to the mucus (Figures 1.6a and 1.6b). At this stage, the cilia may push upwards on the mucus, raising it a little away from the cell surface. As the effective stroke proceeds, the cilium engages with the mucus and imparts propulsive force to the mucus. Towards the end of the effective stroke, the ciliary tip must begin to swing downwards once more and its forward movement decreases. The mucus continues to move forward as it is propelled by surrounding cilia and is pulled away from the decelerating ciliary tip. Then, decelerating cilia finish at the rest position. The resting cilium projects in the direction of mucus transport, and an area of resting cilia could act as a "nonreturn surface", discouraging a backward flow of mucus over the area (Figure 1

.6d).

A cilium that is propelling mucus is subjected to a different distribution of forces from that of a cilium propelling water. In a homogeneous fluid, the viscous resistance to movement of a cilium increases towards the tip only because of increasing velocity of movement. However, a mucus-propelling cilium in its effective stroke is surrounded by low viscosity fluid throughout most of its length, but meets the much greater resistance of mucus in the distal 0.5 µm or so at the tip. Propulsive force for ciliary movement is generated throughout the ciliary length, and when a cilium meets resistance it will tend to bend backwards, the backward bend increasing as the distance of the effective resistance from the ciliary base increases. If a cilium bends backwards too far it will no longer be able to propel mucus effectively (Figure 1.6c). That is why there should be an optimal length for a mucus-propelling cilium at which it can exert a reasonable force and yet not bend too much when it meets a mucus load [START_REF] Sleigh | Movement and coordination of tracheal cilia and the relation of these to mucus transport[END_REF]. Cilia that propel mucus do not work alone but as members of a metachronal wave. They take advantage of the recovery stroke to coordinate. If two cilia lie close enough together, they will interfere with one another and will adjust their frequency and phase of beating so as to minimize this interference [START_REF] Sleigh | The biology of cilia and flagella[END_REF]. According to the positional relationship between them relative to their plane of beating, they may end up beating in phase with one another, or with a constant phase difference. When many cilia interfere with their neighbors in such a way, their beating will become organized into coordinated metachronal waves (see Figure 1.7). We can imagine that changes in the viscosity of the medium, or in the length, or in the spacing of the cilia, will have a deep influence on the characteristics of the metachronal wave. Its wavelength is of the order of a hundred of cilia, namely around 30 µm (see Table 1.1).

Even a small raft of flake of mucus would be likely to be propelled by several metachronal waves. A cilium performing its effective stroke will therefore normally come into contact with mucus that is already moving at perhaps 200 µm.s ´1 [START_REF] Wanner | Alteration of tracheal mucociliary transport in airway disease : effect of pharmacologic agents[END_REF]. Because the ciliary tip would move at 4 or 5 times this speed in the absence of mucus, it will exert a force on the mucus tending to accelerate the mucus. However, the highly viscous nature of the mucus will resist the propulsive force of the cilium and as a consequence will slow down the local speed of the cilium. Other cilia of the metachronal wave moving behind will join the first cilium and add their propulsive effort, and the slower the mucus moves, the larger the number of cilia within each metachronal wave that contribute effective strokes at any instant because they are slowed down to the speed of the mucus.

Based on the works of Sanderson and Sleigh [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF] on the rabbit, waves may be small circular patches of activity, as in Figures 1.8c and 1.8d, or longer areas as seen in Figures 1.8a, 1.8b and 1.8e. In either case, they are limited in space and propagated for only a short distance on the cultured epithelium. In Figures 1.8c and 1.8d, some cilia are performing a clockwise recovery stroke (r), others an effective stroke (e) in the direction of mucus transport. The resultant metachronal wave (m) moves in a different direction.

The pattern of ciliary activity represented in Figure 1.9 indicates the metachronal relationships of the component cilia at an instant during the extension and propagation of the wave across the epithelium. The metachronal wave has originated from a single cilium. Each cilium begins its movement with a clockwise recovery stroke that has two components of hydrodynamic coupling which induce the movement of neighbouring cilia. With respect to the direction of the effective stroke (e) (which is the direction of the mucus transport), this coupling by sideways and backward forces, acting respectively downwards and to the left, as indicated by the short arrow (m). The effective stroke returns the cilia to the resting position by moving in the direction of the arrow (e) and has little influence on the metachronism. The metachronal wave therefore travels at an obtuse angle (" 135

˝)
to the effective stroke. The line of synchrony (s) is at 90 ˝to the direction of the main line of metachrony.

Since there can be several metachronal waves, even if they can be located on small circular patches of ciliary activity (Figure 1.8b), their contribution to the propulsion of an area of mucus will carry the mucus over any irregularities or nonciliated areas of the epithelium, and smooth out any minor differences in rate or direction of beat of the small metachronal fields that go to make up the whole surface. These works on the rabbit were confirmed by Marino and Aiello [START_REF] Marino | Cinemicrographic analysis of beat dynamics of human respiratory cilia[END_REF], whose studies on human bronchial biopsies showed similar restricted metachronal fields. These are interpreted as being limited by the irregularities of the epithelial surface, or probably narrow nonciliated areas or other discontinuities across which mechanical communication of the wave does not easily take place. 1.1) and each one is represented by a single numeral, whose value denotes the phase of the cilium in its beat cycle (corresponding to schema a). Modified from [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF].

Finally, the cilia are very slender bodies whose beats propel the mucus. They move in the periciliary layer, a low viscosity layer beneath the mucus. The beat of a cilium is a liking of a recovery stroke and an effective stroke, interspersed with rest phases, during which the cilium projects in the direction of mucus transport. Only tips of the cilia enter the mucus during the effective stroke to propel it. The cilium does not work alone but as member of a metachronal wave which emerges from the coordination of the cilia during the recovery stroke, and propagates in the opposite direction of mucus transport. All the data related to the cilia and the metachronal wave are given in Table 1.1.

Some pathologies

Before tackling the mathematical modellings for the mucociliary transport, let us mention some pathologies related to it. First, the cilia of persons who inherit the various forms of primary ciliary dyskinesia (immotile cilia syndrome) or Young's syndrome are either totally nonmotile or have such defective motility that mucociliary clearance is absent or severely reduced [START_REF] Camner | Measurements of tracheobronchial clearance in patients with immotile-cilia syndrome and its value in differential diagnosis[END_REF][START_REF] Pavia | Lung mucociliary clearance in patients with Young's syndrome[END_REF][START_REF] Pedersen | Bronchopulmonary symptoms in primary ciliary dyskinesia. A clinical study of 27 patients[END_REF]. In asthma [START_REF] Laitinen | Damage of the airway epithelium and bronchial reactivity in patients with asthma[END_REF], in infections with influenza [START_REF] Camner | Tracheobronchial clearance in patients with influenza[END_REF] or cold viruses [START_REF] Pedersen | Nasal mucociliary transport, number of ciliated cells, and beating pattern in naturally acquired common colds[END_REF] and mycoplasmas [START_REF] Collier | Relationships between mycoplasma pneumoniae and human respiratory epithelium[END_REF], among others, substantial numbers of ciliated cells may be shed, and the propulsive machinery of mucociliary transport depleted. Disorientation of ciliated cells may occur in regeneration after infections, and it appears to be common in bronchitics [START_REF] Iravani | Mucus transport in the tracheobronchial tree of normal and bronchitic rats[END_REF], asthmatics, and smokers [START_REF] Sleigh | Ciliary activity in the respiratory tract and the effects of tobacco smoke[END_REF]. Hypertrophy of the mucous glands is a standard feature of disease, also seen most often in bronchitics [START_REF] Hogg | Site and nature of airway obstruction in chronic obstructive lung disease[END_REF], asthmatics [START_REF] Wanner | Clinical aspects of mucociliary transport[END_REF], and smokers, and hypersecretion of mucus increases the amount to be cleared, perhaps by 10 or 20 times. Excessive mucin secretion by different secretory cell types is likely to be a major cause of the airway plugging associated with fatal asthma [START_REF] Groneberg | Mucus and fatal asthma[END_REF]. Finally, mucus viscosity can change too and disturb mucociliary clearance: it increases in diseases such as cystic fibrosis or when the mucosa becomes dehydrated, for example with a fever or excessive alcohol consumption; it decreases under emotional stress or if the subject has bronchitis or asthma.

Mucociliary transport, state of art

While numerous physiological and pathophysiological factors are known to influence mucociliary clearance and associated diseases, there are often insufficient data to evaluate each of their effects on the mucus transport [START_REF] Del Donno | The effect of inflammation on mucociliary clearance in asthma : An overview[END_REF], motivating theoretical and experimental modelling studies. Even if this phenomenon has already been widely studied, it is still a challenge for the scientific research. Several problems emerge from the complexity of the system: taking into account active thin strutures (cilia), the non-constant viscosity, the Newtonian or non-Newtonian behavior of the fluid, the interface between the two layers, the lining of the bronchi on the periciliary layer, the influence of the air on the mucus, the effects of the inhaled pathogens, etc. This complexity makes the full numerical problem be out of reach of scientists [START_REF] Causey | Mucus and the mare : how little we know[END_REF], but several complementary models have been developed to study the mucociliary clearance. This section is dedicated to the presentation of the literature related to the mucociliary transport in the lung. Mathematical models have been increasingly sophisticated and reflect many of the features found in the actual system, incorporating the effects of a large number of cilia together with the two-layer characteristic of the mucociliary fluid. Several of the more successful mucociliary transport models that have been developped are discussed below.

Envelope models

The concept of the envelope model is based on the idea that the main propulsive thrust comes from the instantaneous surface covering the tips of the undulating cilia (Figure 1.10). This idea was developed independently by Ross and Corrsin [START_REF] Ross | Results of an analytical model of mucociliary pumping[END_REF] for mucus transport and by Blake [START_REF] Blake | Infinite models for ciliary propulsion[END_REF][START_REF] Blake | A spherical envelope approach to ciliary propulsion[END_REF] for free-swimming ciliated protozoa. The mucus transport model of Ross and Corrsin considered two layers: a "watery" periciliary layer and a linear viscoelastic mucous layer (Maxwell fluid). They predicted unrealistically low mucus transport rates, but their theory did not allow contact between the envelope and the viscoelastic mucous layer. This work suggested that the mucous layer moves essentially like an elastic bloc. Moderate changes in the mucus viscosity did not appear to significantly alter transport. Conversely, they noted the significance of the viscosity of the periciliary layer, as any change can yield an appreciable change in mucus transport rate. The envelope model, while useful in gaining some understanding of mucociliary transport, is somewhat superficial in that it does not incorporate the essential physics of the ciliary beat cycle and the mechanical contact with highly viscous mucous layer. A more realistic model would take into account the action of the full length of the cilia and their asymetric beat pattern, which allows the inclusion of such important factors as the velocity and orientation of the cilium during the effective and recovery strokes.

Sublayer models

A wide class of sublayer models have been developed taking into account the full length of the cilia to simulate the propulsion of mucus by cilia. On the one hand, there is the "traction-layer" model which uses a continuum representation for the cilia. On the other hand, there is the "discrete-cilia" model which employs an array of oscillating needlelike bodies to represent the cilia.

Traction-layer model

The action of a large number of cilia is modeled by a continuous distribution of forces per unit volume within the cilia sublayer. The volume force distribution is obtained by appropriate spatial averaging and Fourier analysis over a cycle of a particular ciliary beat pattern. In the model of Keller and coworkers [START_REF] Keller | A traction layer model for ciliary propulsion[END_REF], the averaging technique does not adequately discriminate between the effective and recovery strokes and predicts relatively high oscillatory velocities as a consequence. A fairly constant stream flow in the upper part of the cilia sublayer could be observed. One of the characteristic results of this analysis is the almost total retardation of flow in the lower part of the cilia sublayer together with a "plug flow" in the mucous layer.

Recently Smith and coworkers [START_REF] Smith | A viscoelastic traction layer model of muco-ciliary transport[END_REF] modeled in two-dimension the airway surface liquid by a three-layer model: the upper (the mucus) modeled as a Maxwell fluid, the lower one (the periciliary layer) and the middle one (kind of transition layer, rather thin) considered as Newtonian fluids. The cilia are treated as an active porus medium and their effects modeled by a volumique resistive force depending on the local velovity of the cilia. This model showed a quasi-uniform transport of the mucus layer but no collective movement in the periciliary layer emerges from this model.

Discrete-cilia model

Barton and Raynor [START_REF] Barton | Analytical investigation of cilia induced mucous flow[END_REF] developed a mathematical analysis of ciliary propulsion, approximating the cilium by a rigid rod which automatically shortens during the recovery stroke. They used a simplified "resistance coefficient" approximation to the effect of the cilium on the surrounding fluid, and their model of the cilia motion did not include the experimental findings that were to emerge later. Their work allowed us to calculate realistic flow rates, and their characterization of the cilium as a rigid rod was exploited later in some models [START_REF] Blake | On the mechanics of muco-ciliary transport[END_REF].

An alternate "discrete sublayer" approach utilizing the mathematical technique of slender-body theory was developed by Blake [START_REF] Blake | A model for the micro-structure in ciliated organisms[END_REF], initially for ciliated microorganisms. Due to their slenderness, individual cilia can be modelled by distributing force singularities along their centrelines. These ideas were extended in studies such as Liron and Mochon [START_REF] Liron | The discrete-cilia approach to propulsion of ciliated microorganisms[END_REF] and Fulford and Blake [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF]. This theoretical approach gives estimates for the mean field velocity in both the periciliary layer and the mucous layer when both liquids are assumed to have Newtonian fluid properties. In all examples, the mean field velocity is very small in the lower part of the periciliary layer (which contains the recovery stroke) and increases very quickly close to the mucous layer, up to the mucous velocity.

Cilia seem to alter their beat pattern in the presence of mucus, or in a general way when subjected to a high viscosity environment. Calculations using a specified beat pattern do not strongly support the hypothesis that the penetration of the cilia into the mucous layer during their effective stroke is essential for mucus movement. Actually, they suggest that ample velocities of propulsion are obtained by the cilia even if there is no penetration, in the situation where the cilia densities are sufficiently high. Conversely, calculations based on models in which active cilia are sparsely distributed over the epithelium suggest that penetration of the cilia into the mucous layer would significantly enhance transport.

The form of the ciliary beating, especially during its effective stroke, is a determining factor for the velocity of the mucous layer. The ciliary beat described by Sanderson and Sleigh [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF] is the more realistic as the cilia encounter the very large resistance of the mucus. Other theories have been developed, they include the tip penetration models, in which only the tips of the cilia penetrate the mucus [START_REF] Liron | Muco-ciliary transport[END_REF], and the difficulties to determine the deformation of the interface between two viscous fluids by the motion of a nearby slender body [START_REF] Blake | Mechanics of ciliary transport[END_REF].

More recently, Smith and coworkers [START_REF] Smith | Discrete cilia modelling with singularity distributions : application to the embryonic node and the airway surface liquid[END_REF] discussed in detail techniques for modelling flows due to finite and infinite arrays of beating cilia. Cilia are modelled as curved slender ellipsoidal bodies by distributing Stokeslet and potential source dipole singularities along their centrelines, leading to an integral equation that can be solved using a simple and efficient discretization. They developed a more complex and detailed model of flow patterns in the periciliary layer of the airway surface liquid. Their results confirmed that shear flow of the mucus layer drives a significant volume of periciliary liquid in the direction of mucus transport even during the recovery stroke of the cilia. Lee and coworkers [START_REF] Lee | Muco-ciliary transport : effect of mucus viscosity, cilia beat frequency and cilia density[END_REF] considered in 2-dimension a two-layer Newtonian fluid model to investigate the important factors that may affect the mucociliary transport. The cilium is modeled by discrete Dirac delta function distributed all along the cilium. The numerical techinque implemented in this study is a projection method combined with the immersed boundary method in order to prescribe the ciliary beating patterns. Using the same computational techniques, Jayathilake and coworkers [START_REF] Jayathilake | Three-dimensional numerical simulations of human pulmonary cilia in the periciliary liquid layer by the immersed boundary method[END_REF] simulated a 3-dimensional two-layer flow to better understand some pathological cases.

Non constant viscosity models

A few works consider a variable-viscosity model. Matar and Spelt [START_REF] Matar | Dynamics of thin free films with reaction-driven density and viscosity variations[END_REF] modeled in 2dimension the viscosity as the solution of a reaction-advection-diffusion equation depending on the temperature, but the model of the flow is very simplified. A variable-viscosity model has been presented by Enault and coworkers [START_REF] Enault | Mucus dynamics subject to air and wall motion[END_REF] in the abscence of cilia, and by Mauroy and coworkers [START_REF] Mauroy | Toward the modeling of mucus draining from the human lung : role of the geometry of the airway tree[END_REF] at the scale of the bronchi for a Bingham fluid. Chatelin and Poncet [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF] proposed a 3-dimensional model where the viscosity is the solution of a convection-diffusion equation. The movement of the cilia is imposed as the solution of an 1-dimensional transport equation on a parametric curve, and the effects of the cilia on the fluid are treated by penalization. To the best of our knowledge, in the other works which consider a variable viscosity, the viscosity is defined constant by part. In a recent study, Dillon and coworkers [START_REF] Dillon | Fluid dynamic models of flagellar and ciliary beating[END_REF] used the immersed boundary method to simulate 2-dimensional flow due to three cilia, with a discrete model of the internal elastic and force-generating structures of the cilia. The mucus layer was simulated using a relatively thin elastic layer immersed in the fluid, a short distance above the cilia tips. Their results showed that the cilia are able to propel mucus without the need to engage with the mucus layer, and that elastic properties of the mucus layer prevent unphysical deformation of the mucuspericiliary liquid interface. This observation justifies the two-layers models (sometimes three) developed in several works [START_REF] Jayathilake | Three-dimensional numerical simulations of human pulmonary cilia in the periciliary liquid layer by the immersed boundary method[END_REF][START_REF] Lee | Muco-ciliary transport : effect of mucus viscosity, cilia beat frequency and cilia density[END_REF][START_REF] Smith | A viscoelastic traction layer model of muco-ciliary transport[END_REF], for which the interfaces are assumed invariable and flat.

For a more complete synthesis of the works on the mucociliary transport, we refer the reader to the review by Smith and coworkers [START_REF] Smith | Modelling mucociliary clearance[END_REF]. An important conclusion of this state of art is that the complexity of the phenomenon leads us to consider several restrictive assumptions. The next section is devoted to the presentation of the mathematical model of mucociliary transport. The aim is to simulate in 3D a whole forest of cilia in a twoviscosity fluid. We will especially pay attention to limit the computational costs related to the model, and at the same time, try to keep assumptions to the minimum.

Mathematical modelling of mucociliary transport

In discussing the fluid mechanical principles of mucociliary transport, it should be observed that there are several length scales of particular relevance to this study. There are:

' molecular length scales relative to the biochemical structure of the mucus.

' length scales associated with the cilium tip (0.1 to 1 µm).

' a length associated with the cilium length, cell size, ciliary wavelength and coherence of a mucous plaque (5 to 50 µm).

' a length scale associated with the length of an airway (5 to 10 mm).

In this study, we focus on the mucociliary transport phenomena at the scale of the cilium and ciliary wavelength.

Modeling of the fluid

A Newtonian fluid

All the notations introduced in the following developement are summarised in Table 1.2. As a first step, the fluid is assumed to be homogenous with constant viscosity. The mathematical modelling of fluid is classically based on the two conservation principles:

' mass conservation for an incompressible and homogeneous fluid,

divpuq " 0 on Ω, ' movement quantity conservation, ρ ˆBu Bt `divpu b uq ˙" divpσq `Fe on Ω.
We assume that the fluid is Newtonian:

σ " 2µDpuq ´pI 3 " µ `∇u `t∇u ˘´pI 3 , (1.3.1)
which leads to the incompressible Navier-Stokes equations:

$ & % ρ ˆBu Bt `divpu b uq ˙´µ△u `∇p " F e on Ω,
divpuq " 0 on Ω.

(1.3.2)

As it has been discussed in Section 1.1.2.1, the mucus is a viscoelastic fluid, and so we should work with a non-Newtonian fluid model. Nevertheless, the relaxation time is very high, which means that at the scale of the cilia beat, the elastic energy of the fluid is insignificant with regard to the viscous resistance. As a consequence, we have chosen to model the mucus as a Newtonian fluid, and use Equation ( 1 where r u, r t, r p and r F e are the corresponding dimensionless quantities, and Re is the Reynolds number. It is defined by

Re " ρLU µ ,
with L and U respectively the flow characteristic length and velocity. Let us discuss two particular regimes:

' either the Reynolds number Re is high, which is true at the scale of the human swimmer (Figure 1.11a), for who Re is of the order of 10 6 . In this case, inertial terms in Equation (1.3.3) preponderate over viscous terms. Note that these equations are not reversible in time because of the non-linear term, and the solution depends on the solution in the past.

' or the Reynolds number Re is low, which is true for micro-swimmers like Chlamydomonas Reinhardtii (Figure 1.11b), for which Re is of the order of 10 ´6. This time, the inertial terms are dominated by the viscous terms and thus, the fluid can be modeled by the Stokes equations " ´µ△u `∇p " F e on Ω, divpuq " 0 on Ω.

(1.3.4)

Let us note that the time-derivative does not appear in the Stokes equations, which results in an instantaneous balance of forces at each time: there is no dependance on the past, no memory effects, and the problem is completely reversible in time.

Regarding to mucociliary transport, the characteristic length and velocity are of the order of the micrometer and the micrometer per second, so that the Reynolds number Re in this case is of the order of 10 ´6, like the micro-swimmer Chlamydomonas Reinhardtii. As a consequence, the Stokes equations are privileged. It is very important to underline that, because of the reversibility of the Stokes equations, the mucous transport is efficient only if the movement of the cilia is not reversible in time.

Much of the mucus propelled by cilia in the tracheobronchial tree moves against the pull of gravity, even if values of mucociliary transport appear similar irrespective of orientation. At the micrometric scale, the force of gravity ρg and of buoyancy △ρg are insignificant. Moreover, for normal mucus depths (ď 10 µm), two factors negate the gravitational effects: first, the high-density array of cilia acting like a porous resistive medium, and second, the high viscosity and tensile forces within the mucus itself. The conclusion is that there is no external forces: F e " 0. ' the easiest model is the "one-way" model, that is only one action is considered. Here, we model the effects of the cilia on the fluid, but we neglect the retroaction of the fluid on the cilia. In this case, the positions and the velocities of the cilia are given either by solving some mechanics equations or analytically imposed.

' the second model, more complex, is the "two-ways" model, when the two interactions are considered. The mechanical equations are coupled with the fluid mechanics equations (for example through the boundary conditions) and both the action of the cilia on the fluid and the action of the fluid on the cilia are modeled, leading to a strongly coupled fluid-structure interaction problem.

In this work, we consider only the action of the cilia on the mucus ("one-way" model) and we will see in Section 1.3. As described in Section 1.1.2, the airway surface liquid is composed of two overlayed layers, the mucus and the periciliary layer. In our model, we take into account this fact

" µ 1 if z ď h 0 , µ 2 if z ě h 0 ,
where h 0 is the height of the periciliary layer. Finally, Problem (1. For now, we impose one boundary condition: u " u cil on the cilia (no-slip boundary conditions). In this section, we describe the different boundary conditions we impose to match reality. First, consider a box as the domain Ω (see Figure 1.13). The bottom corresponds to the lining of the bronchi, which is assumed to be flat, and we impose no-slip boundary condition: u " 0. The box we consider is seen as a window focused on a part of a bronchus, with mucociliary transport which goes on outside the box. Therefore it is natural to impose in both directions x and y biperiodic boundary conditions. Lastly, if we pay attention to Figure 1.7, it seems reasonable to assume that the mucus surface remains flat. This approximation involves two boundary conditions on the top of the box: " u ¨n " 0 kinematic condition, pσ ¨nq ¨τ " 0 dynamic condition,

where n is the normal vertor, τ the tangential vector and σ the constraint tensor. The kinematic condition corresponds to a non-output condition while the dynamic condition is a no-friction sliding condition. All the boundary conditions we impose in our model are summarized in Figure 1.13.

For the theoretical study, we do not consider these boundary conditons. Actually, they do not impact on the regularity of the solution and/or the approximation of the solution by finite element methods. Thus, we will consider homogeneous Dirichlet boundary conditions fir the sake of clarity, even if the results readily adapt with the boundary conditions that have been described in this section. They will be considered at the time of the computations (Chapter 4). In the meantime, Problem (1.3.6) is well-defined and we can discuss the regularity of the solution.

Modeling of a cilium

Hierarchy of thin-structure models for the cilium

The aim of this work is to solve Problem (1.3.6) by a finite-element method, so we have to mesh the domain Ω correctly. In the previously chosen model, in order to take into account the cilia in a conforming way, the mesh has to respect the complex geometry of domain Ω, especially at the neighbourhood of the cilia. The cilia are very slender bodies, as a consequence the mesh has to be very refined near the cilia. These considerations mean that computational costs will be very high, not to mention cilia beat very quickly and the mesh has to be redefined at each time step. If we could imagine this process for one cilium, it is clearly not adapted to a forest of cilia.

Another way to treat the cilia is to process by penalisation: the domain Ω includes the cilia, and the velocity of the fluid is penalized to be equal to the cilium velocity. Once more, it is important to have a very refined mesh on the volume occupied by the cilia; otherwise, cilia seen by the mesh would behave as a continuous layer where the velocity of the fluid is imposed (this model tends to the traction-layer model). Finally, this model yields the same computational cost issues as the previous model, and therefore is not adapted neither.

Our model is based on the fact that the cilium is a very slender body (the ratio crosssectional radius over length is r 0 {L " 0.1{6) and that it beats very quickly (frenquence f " 15 Hz), see Table 1.1. To avoid too prohibitive computational costs, we consider the asymptotic of a zero diameter cilium with an infinite velocity: the cilium is modeled by a line distribution of forces in source term. Again, in order to ease the computations, the line distribution of forces is approached by a sum of punctual forces distributed along the cilium, Figure 1.14.

In this section, we estimate the convergences between these different source terms: the volumic source term, the line distribution of forces and the distribution of punctual forces. More precisely, see Figure 1.14, a. the cilium seen as a curved cylinder, which corresponds to a volumic source term of radius ε ą 0, b. the cilium seen as a line distribution of forces on the curve Γ,

f ε " 1 πε 2 1 cil P L 2 pΩq,
δ Γ P H ´1´η pΩq, @η ą 0, (1.3.7)
and u Γ P H 1´η pΩq the associated solution.

c. the previous line distribution replaced with a sum of N punctual forces located at the points x i and of intensities c i ,

f N " N ÿ i"1 c i δ x i P H ´3{2´η pΩq, @η ą 0,
and u N P H 1{2´η pΩq the associated solution. Let us define ξ : r0, Ls Ñ R 3 s Þ Ñ ξpsq a regular parametrization of Γ such that ξ P C 8 and d s ξ is never zero. We recall the following properties of the Sobolev spaces:

ÝÝÝÝÝÑ εÑ0 ÐÝÝÝÝÝ N Ñ`8 2ε Ð Ñ δ Γ N ÿ i"1 c i δ x i a. b. c.
Proposition 1 (Sobolev embeddings). Consider a regular domain Ω, we denote by d the dimension,

H m pΩq ãÑ C k p Ωq for m ą k `d 2 .
Proposition 2 (Trace operator). Consider s ą 1{2 and an open domain Ω with a regular border BΩ, there exists a continuous mapping T defined by

T : H s pΩq ÝÑ H s´1{2 pΩq v Þ ÝÑ T v " v | BΩ .
We can now establish estimates of convergence between these different source terms and deduce convergences for the solutions. From now and until the end of this section, the study is led in dimension 3. We denote by η any (small) real such as η ą 0.

Convergence of f N to δ Γ . We show a weak convergence result in H 3{2`η pΩq and a strong convergence result in H 2`η pΩq.

Proposition 3 (Weak convergence). Let px i q iPrr1,N ss be N orderly points on Γ, there exist

N reals pc i q iPrr1,N ss such that, for all v P H 3{2`η 0 pΩq, x N ÿ i"1 c x i δ x i , vy ÝÑ N Ñ`8 xδ Γ , vy, which means that f N " ř c x i δ x i weakly converges to δ Γ in H ´3{2´η .
Proof. Consider v P H 3{2`η . By Proposition 1, v is continuous, so that we can write:

x N ÿ i"1 c x i δ x i , vy " N ÿ i"1 c x i vpx i q.
Moreover, for all i, there exists s i P r0, Ls such that x i " ξps i q, so we have

N ÿ i"1 c x i vpx i q " N ÿ i"1 c ξps i q vpξps i qq.
On the other hand,

xδ Γ , vy " ż Γ vpxqdx " ż L 0 vpξpsqq › › › › dξ ds psq › › › › ds.
So, let us choose ps i q i such as

0 " s 0 ă s 1 ă ¨¨¨ă s N " L et max i s i ´si´1 À L N , and
c ξps i q " › › › › dξ ds ps i q › › › › ps i ´si´1 q.
Since the product › › dξ ds p¨q › › vpξp¨qq is continuous, By the Riemann sums theorem,

N ÿ i"1 ps i ´si´1 q › › › › dξ ds ps i q › › › › vpξps i qq ÝÑ N Ñ`8 ż L 0 › › › › dξ ds psq › › › › vpξpsqqds " ż Γ vpxqdx.
Therefore, for all v P H 3{2`η 0 pΩq,

x

N ÿ i"1 c x i δ x i , vy ÝÑ N Ñ`8 xδ Γ , vy,
which ends the proof of the weak converge.

Proposition 4 (Strong convergence). The convergence rate of f N " ř c x i δ x i to δ Γ depends on the space we consider:

1.3. Mathematical modelling of mucociliary transport 59 ' In H ´2´η pΩq, f N converges to δ Γ at the order N ´1{2 . ' In H ´5{2´η pΩq, f N converges to δ Γ at the order N ´1. Proof. Consider v P H 2`η pΩq, formally, ˇˇˇˇx N ÿ i"1 c x i δ x i , vy ´xδ Γ , vy ˇˇˇˇ" ˇˇˇˇN ÿ i"1 ps i ´si´1 q › › › › dξ ds ps i q › › › › vpξps i qq ´ż L 0 › › › › dξ ds psq › › › › vpξpsqqds ˇˇˇď N ÿ i"1 ż s i s i´1 ˇˇˇ› › › › dξ ds ps i q › › › › vpξps i qq ´› › › › dξ ds psq › › › › vpξpsqq ˇˇˇd s (1.3.8) ď N ÿ i"1 ż s i s i´1 ˇˇˇż s i s B s ˆ› › › › dξ ds p¨q › › › › vpξp¨qq ˙pσqdσ ˇˇˇd s (1.3.9) ď N ÿ i"1 ż s i s i´1 ps i ´si´1 q 1{2 › › › › B s ˆ› › › › dξ ds p¨q › › › › vpξp¨qq ˙› › › › H η ps0,Lrq ds (1.3.10) ď Cpξq}v} H 2`η pΩq ˆL N ˙1{2 .
Let us derive this estimate in a rigorous way:

' from (1.3.8) to (1.3.9), since v P H 2`η pΩq, the trace operator which restricts v to Γ defines a function r v P H 1`η pΓq, and thus the composition by the regular function ξ becomes a function of H 1`η ps0, Lrq. So we can apply the fundamental theorem of calculus which writes a function as the integral of its derivative (mutliplying by the regular function } dξ ds p¨q} does not perturb this process).

' Inequality (1.3.10) comes from the Cauchy-Schwarz inequality.

Thus, we have shown that f N " ř c x i δ x i converges to δ Γ in H ´2´η pΩq at the order N ´1{2 . Note that this result is a priori false in H ´3{2´η pΩq.

We can also specify the convergence rate in H ´5{2´η . Indeed, Riemann sums theorem ensures that the rate of convergence for C 1 pΩq-functions is OpN ´1q. More precisely,

ˇˇˇˇx N ÿ i"1 c x i δ x i , vy ´xδ Γ , vy ˇˇˇˇď Cpξq L N }∇v} 8, Ω.
By Proposition 1, for any v P H 5{2`η , }∇v} 8, Ω ď C}v} H 5{2`η pΩq . We conclude that f N " ř c x i δ x i converges to δ Γ in H ´5{2´η pΩq at the order N ´1.

Chapter 1. From the mucociliary transport to the Stokes problem Convergence of f ε to δ Γ By the definition of δ Γ , Equation (1.3.7), the test function v has to belong to H 1`η pΩq, so that xδ Γ , vy is well-defined (actually, we need that the trace operator is defined on a submanifold of dimension d ´2). Nevertheless, conversely to the previous case, no convergence (even weak) has been established in H ´1´η pΩq. We can however get an order of convergence in a weaker norm.

Proposition 5. The convergence rate of f ε to δ Γ depends on the space we consider:

' In H ´2´η pΩq, f ε strongly converges to δ Γ at the order ? ε.

' In H ´5{2´η pΩq, f ε strongly converges to δ Γ at the order ε.

Proof. For v P H where Dpy, εq is the cross-section of the cylinder C ε at the point y P Γ. We conclude that f ε converges to δ Γ at the order 1{2 in H ´2´η pΩq.

We can also specify the convergence in H ´5{2´η pΩq. Indeed, for v P H 5{2`η pΩq, from the above,

|xf ε , vy ´xδ Γ , vy| " ˇˇˇż Cε 1 πε 2 ´vpxq ´vpp Γ pxqq ¯dx ˇˇď ż Cε 1 πε 2 }∇v} 8,Ω |x ´pΓ pxq|dx ď ε|Γ|}∇v} 8,Ω .
We conclude with Proposition 1 that f ε converges to δ Γ at the order 1 in H ´5{2´η pΩq.

Mathematical modelling of mucociliary transport 61

Convergence of the solutions. Now we have established convergence results for the source terms, we can specify the convergence results for the corresponding solutions. The results of this section are based on the following theorem, taken from [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] (Chapter 2, Theorem 6.3).

Theorem 1. Define the space

Ξ r " $ & % v P L 2 pΩq ˇˇˇ} v} Ξ r " ÿ |α|ďr }ρ |α| D α v} 0 ă `8, .
-

,
where ρ is a smooth function such that there exist c 1 , c 2 P R ˚satisfying

c 1 ď ρpxq dpx, BΩq ď c 2 , @x P Ω.
The space Ξ ´r, for r ě 0, is defined as the dual space of Ξ r . Consider the following elliptic problem

" Au " f in Ω, u " 0 on BΩ, (1.3.11)
where A is a "reasonable" elliptic operator (we assume that (1.3.11) has a unique solution for any regular right-hand side f ). Then, for any f in Ξ ´r, r ě 0, there exists a solution u of (1.3.11) satisfying

}u} m´r ď C}f } Ξ ´r ,
where m is the highest order of derivation in the operator A.

Inspired by Scott [START_REF] Scott | Finite element convergence for singular data[END_REF], we apply Theorem 1 to δ Γ ´ř c x i δ x i and f ε ´δΓ . But in order to do so we need to assume the cilium to be "far" from the border of Ω, that is, there exists Ω 0 ĂĂ Ω such as supports of f ε , δ Γ and ř c x i δ x i are included in Ω 0 . Indeed, the H ´r-norm and the Ξ ´r-norm are equivalent on Ω 0 , and for any f supported in Ω 0 , we have }f } Ξ ´r ď cdpΩ 0 , BΩq ´r}f } ´r, where c depends only on Ω 0 and on the choice of ρ. Finally, we conclude easily from the previous sections that for any η ą 0,

}u Γ ´uN } ´η ď C › › › › › δ Γ ´N ÿ c x i δ x i › › › › › Ξ ´2´η ď C › › › › › δ Γ ´N ÿ c x i δ x i › › › › › ´2´η ď C ? N , }u Γ ´uN } ´1{2´η ď C › › › › › δ Γ ´N ÿ c x i δ x i › › › › › Ξ ´5{2´η ď C › › › › › δ Γ ´N ÿ c x i δ x i › › › › › ´5{2´η ď C N , }u ε ´uΓ } ´η ď C}f ε ´δΓ } Ξ ´2´η ď C}f ε ´δΓ } ´2´η ď C ? ε, }u ε ´uΓ } ´1{2´η ď C}f ε ´δΓ } Ξ ´5{2´η ď C}f ε ´δΓ } ´5{2´η ď Cε.
Remark 1. The convergence of the solutions in weak spaces (relative to the solution's regularities) has been established under the assumption "the cilium is far from the border".

In the case of mucociliary transport, the cilium is connected to the lining of the bronchi, which is the border of our domain Ω. Nevertheless, the velocity of the cilium at the border is zero, and so the active part of the cilium is not in contact with the border: the convergence of the solutions holds in this case.

Parametrization of a cilium and of a forest of cilia

As explained in Sections 1.3.1.3 and 1.3.2.1, the effect of the cilia on the fluid ("one-way" model) is modeled by punctual forces distributed along the cilia. To complete this model, we have to define a parametrization of the movement of each cilium and the force it exerts on the fluid. Let us start with the parametrization of the movement of a cilium.

One cilium. Conversely to inert stiff or elastic solids, the cilia are "active": they have their own internal energy which allows them to beat. In our model, their movement is analytically imposed, as it is done for instance by Chatelin and Poncet in [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF]. We use the parametrization established by Fulford and Blake in [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF], based on the Fourier series decomposition of the beat of the cilium described by Sanderson and Sleigh in [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF]. More precisely, at each time t the cilium is represented by the truncated Fourier series of the parametric curve

ξps, tq " L « 1 2 a 0 psq `6 ÿ n"1
a n psq cosp2nπf tq `bn psq sinp2nπf tq ff , where s P r0, 1s measures arclength from the base of the cilium, L and f are respectively the length and the beat frequency of the cilium (see Table 1.1). The Fourier coefficients a n , b n are vector quantities, which are approximated by the following 3-degree polynomial functions

a n psq " 3 ÿ k"1 a n,k s k and b n psq " 3 ÿ k"1 b n,k s k
where a n,k and b n,k are constant vectors of R 3 , given in Table 1.3.

Figure 1.15 shows the beat of a cilium that we get with this parametrization and confirms the correspondance with the description made by Sanderson and Seligh in [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF]. Let us note that the cilium crosses the interface between the mucus and the peryciliary layer (PCL) during the effective stroke (and only during this phase). This point is very important to guarantee an efficient mucus transport. In [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF], Fulford and Blake not only give a parametrization of one cilium, but they also explain how to parametrize a whole forest of cilia.

a n,k n " 0 n " 1 n " 2 n " 3 n " 4 n " 5 n " 6 k " 1 ´0.
A forest of cilia. Now that we have described the beat of one cilium, parametrizing a whole forest is not more complicated if we take into account two important features of the forest: the space ℓ 0 between two cilia (in each direction) and the wavelength λ of the metachronal wave. These data are given in Table 1.1. More precisely, in order to model a forest of n x ˆny cilia, with n x (respectively n y ) the number of cilia in the direction x (respectively the direction y), the cilium pi, jq, where i P rr1, n x ss and j P rr1, n y ss, is Chapter 1. From the mucociliary transport to the Stokes problem parametrized by the curve ξ i,j ps, tq " iℓ 0 e x `jℓ 0 e y `ξ ˆs, iℓ 0 f λ `t˙, where s P r0, 1s. Let us note the phase shift ϕ i " iℓ 0 {f λ in x (and only in x, not in y) which the metachronal wave comes from. As most of the models, we only consider the metachronal wave in the direction x, even if Figure 1.9 shows that activity is also propagated in the direction y. This approximation is classical and experiments [START_REF] Gheber | Effect of viscosity on metachrony in mucus propelling cilia[END_REF] show that increasing viscosity causes the metachronal wave to become more orthoplectic (in the direction of mucociliary transport). We have drawn in Figure 1.16 a section of the forest in the direction x. The propagation of the metachronal wave (to the left on the picture) is in the opposite direction of mucus transport (to the right). Figure 1.17 shows a forest in 3d as we model it in the simulations (see Chapter 4), except on the value of the cilia spacing ℓ 0 : it has been voluntarily increased for the sake of clarity. This section has been dedicated to the parametrization of the cilium and of a forest of cilia. Next section presents the model we have chosen for the force distributed along a cilium.

.2.3 Distribution of forces along the cilium

There are two main theories to model the distribution of forces along a cilium. The simplier one is the resistive-force theory, and the second one is the slender-body theory. The resistive-force theory has been developed for flagella hydrodynamics in the pioneering work of Gray and Hancock [START_REF] Gray | The propulsion of sea-urchin spermatozoa[END_REF]. It has been extensively used in subsequent studies of flagellar propulsion and bending [START_REF] Blum | Biophysics of flagellar motility[END_REF][START_REF] Brokaw | Non-sinusoidal bending waves of sperm flagella[END_REF][START_REF] Brokaw | Bending moments in free-swimming flagella[END_REF]. The underlying assumption of this model is that the hydrodynamic forces are proportional to the local body velocity, where the constant of proportionality is defined as the force (or drag) coefficient. As pointed out by Lighthill [START_REF] Lighthill | Flagellar hydrodynamics[END_REF], this assumption is inconsistent with the true hydrodynamic situation in which viscous effects dominate and produce long-range hydrodynamic interactions. Therefore, we have chosen to use the slender-body theory, although it requires more extensive numerical computations than the resistive-force theory. The slender-body theory model we use is based on asymptotic expansions when the ratio L{r 0 tends to infinity, where r 0 is the thickness of the cilium and L its length. Cox [START_REF] Cox | The motion of long slender bodies in a viscous fluid. Part 1. General theory[END_REF] established an asymptotic expansion of the force at each point of the cilium. If ξ is a parametrization of the cilium, the expression of the force at the point of curvilinear abscissa s is fps, tq " 2πµ lnpL{r 0 q ˜2I 3 ´9 ξps, tq b 9 ξps, tq } 9 ξps, tq} 2 ¸ucil ps, tq,

(1.3.12)

where 9 ξ is the derivative of ξ relative to s and u cil is the derivative relative to t: 9 ξps, tq " Bξ Bs ps, tq and u cil ps, tq " Bξ Bt ps, tq.

Equation (1.3.12) has been established in [START_REF] Cox | The motion of long slender bodies in a viscous fluid. Part 1. General theory[END_REF] by confronting two different approaches:

a. the inner expansion consists in studying the fluid velocity near the cilium, which is thus seen as an infinite cylinder. The corresponding regime is L tends to infinity while r 0 remains constant.

b. the outer expansion considers the flow far from the cilium seen as a zero-thickness body, which corresponds to the regime r 0 goes to zero while L remains constant.

Both of these expansions are illustrated in Figure 1.18.

In the case of a two-viscosity fluid, the slender-body theory described by Cox [START_REF] Cox | The motion of long slender bodies in a viscous fluid. Part 1. General theory[END_REF] is not valid any longer. Fulford and Blake [START_REF] Fulford | Force distribution along a slender body straddling an interface[END_REF] established the expression of the distribution of forces along a slender body which straddles an interface. At the first order (in the regime lnpL{r 0 q ´1 tends to zero) the expression of the force is the one given by Equation (1.3.12) with µ the value of the viscosity at the origin point of the force. Actually, that point is not exact: the expression of the distribution of forces on the part of the cilium close to the interface (at a distance r 0 from the interface) is a little different. In our case, there are 20 punctual forces by cilium, so that the distance between two punctual forces is 0.3 µm, which is three times the radius r 0 " 0.1 µm of the cilium (see Table 1.1). Therefore, we do the following approximation: we consider Equation (1.3.12) for the distribution of forces all along the cilium.

Resulting mathematical problem

This subsection is a conclusion of the whole model presented in Section 1.3. Finally, the model leads us to the following problem

$ ' ' ' ' & ' ' ' ' % ´2divpµDpuqq `∇p " Ncx ÿ i"1 Ncy ÿ j"1 ˜N ÿ n"1 δ ξ i,j psn,tq f ˆsn , iℓ 0 f λ `t˙¸o n Ω, divpuq " 0 on Ω, u " 0 on BΩ, (1.3.13)
where N cx and N cy are respectively the number of cilia in the directions x and y, and N is the number of punctual forces composing each cilium. We recall that ℓ 0 , f , and λ are respectively the distance between two neighbouring cilia, the beat frequency and the wavelength of the metachronal wave, see In what follows, we focus on the regularity of the solution and its approximations by finite element methods. In our model of a two-viscosity fluid, with 0 ă µ 1 ă µ 2 , the regularity of the solution (and so the convergence of the numerical solution) is not altered by the non-constant viscosity. Therefore, we consider the Stokes problem in a constantviscosity fluid. In this case, the study is quite simplier and especially we can make explicit the solutions (at least in an infinite domain). Nevertheless, the two-viscosity fluid will be also considered in Chapter 4 in the computations. Finally, for the sake of clarity, we study the following problem

$ ' & ' % ´µ△u `∇p " δ x 0 F on Ω, divpuq " 0 on Ω, u " 0 on BΩ. (1.3.14)
Before focusing on the numerical method to solve Problem (1.3.14), next section is dedicated to the study of this problem (regularity, existence of solutions...).

Elliptic problems with a singular right-hand side

This section is divided in two parts: the first one is devoted to the study of the Stokes problem with a punctual source term (Problem (1.3.14)); the second one introduces the Poisson problem with a Dirac mass in right-hand side, which is a scalar version of Problem (1.3.14). We take an interest in this problem because it is a simplier problem for a preliminary study, and because many results or methods that apply to the Poisson problem can be adaptated to the Stokes problem.

The Stokes problem with a punctual force in source term

The Stokeslet

The study is lead in 2-dimension, but the results will also be given in dimension 3. For the sake of simplicity, the location of the punctual force is the origin. Let us consider the Stokes problem on the whole domain:

" ´µ△u δ `∇p δ " δ 0 F in R 2 , divpu δ q " 0 in R 2 . (1.4.1)
The solution of this problem is not unique. Indeed, the homogeneous problem

" ´µ△u `∇p " 0 in R 2 , divpuq " 0 in R 2 .
has non-trivial solutions. For instance,

upxq " " cospx 1 qe x 2 sinpx 1 qe x 2
 and p " 0.

In Proposition 6, we consider a particular solution, called "Stokeslet".

Proposition 6. The 2d Stokeslet [START_REF] Chwang | Hydromechanics of low-Reynolds-number flow. II. Singularity method for Stokes flows[END_REF], defined by

u δ pxq " 1 4πµ ˆ´ln }x}I 2 `x ¨tx }x} 2 ˙¨F P R 2 , " 1 4πµ » - - - ´ln }x}f 1 `x2 1 f 1 `x1 x 2 f 2 }x} 2 ´ln }x}f 2 `x1 x 2 f 1 `x2 2 f 2 }x} 2 fi ffi ffi fl P R 2 , p δ pxq " x ¨F 2π}x} 2 " x 1 f 1 `x2 f 2 2π}x} 2 P R, (1.4.2)
where F " t rf 1 , f 2 s and x " t rx 1 , x 2 s, is a solution of Problem (1.4.1).

Proof. Here, we present a constructive proof of the existence of the Stokeslet. As it will be done in the case of the Poisson problem (see Section 1.4.2.1), we first consider the following velocity

u δ pxq " ´c " ln }x}f 1 ln }x}f 2  ,
where c is constant. Defined in this way, u δ satisfies ´△u δ " r cδ 0 , but the condition divpu δ q " 0 is not satisfied. Therefore u δ is modified by

u δ pxq " c " ´ln }x}f 1 `v1 pxqf 1 `v2 pxqf 2 ´ln }x}f 2 `w1 pxqf 1 `w2 pxqf 2  .
Some calculations show that divpu δ q " 0 for any F in R 2 is equivalent to the system

$ ' & ' % B 1 v 1 `B2 w 1 " x 1 }x} 2 B 1 v 2 `B2 w 2 " x 2 }x} 2 .
(1.4.

3)

The right hand side expression of System (1.4.3) encourages us to consider v 1 and w 1 defined by

v 1 pxq " x α 1 1 x β 1 2 }x} 2 et w 1 pxq " x α 2 1 x β 2 2 }x} 2 ,
where α 1 , α 2 , β 1 et β 2 are some reals we have to set. We inject in the first equation of System (1.4.3), and we get the conditions

$ ' ' ' ' & ' ' ' ' % α 1 ´1 " 1 β 1 " 0 α 2 " 1 β 2 ´1 " 0 α 1 `β2 " 3 ùñ $ ' ' & ' ' % α 1 " 2 β 1 " 0 α 2 " 1 β 2 " 1 which gives v 1 pxq " x 2 1 }x} 2 et w 1 pxq " x 1 x 2 }x} 2 .
By injecting in the second equation of (1.4.3), we get

v 2 pxq " x 1 x 2 }x} 2 et w 2 pxq " x 2 2 }x} 2 .
Finally, we have established the expression of the solution u:

u δ pxq " " u 1 pxq u 2 pxq  " c » - - - ´ln }x}f 1 `x2 1 f 1 `x1 x 2 f 2 }x} 2 ´ln }x}f 2 `x1 x 2 f 1 `x2 2 f 2 }x} 2 fi ffi ffi fl .
Now, let us consider the pressure p δ . We want ´µ△u δ `∇p δ " 0 (for x ‰ 0), so we set

∇p δ pxq " cµ△u δ pxq " cµ△ » - - - x 2 1 f 1 `x1 x 2 f 2 }x} 2 x 1 x 2 f 1 `x2 2 f 2 }x} 2 fi ffi ffi fl .
After some calculations, we have

∇p δ pxq " « B 1 p δ pxq B 2 p δ pxq ff " 2cµ » - - - 1 }x} 2 f 1 ´2x 1 }x} 4 px 1 f 1 `x2 f 2 q 1 }x} 2 f 2 ´2x 2 }x} 4 px 1 f 1 `x2 f 2 q fi ffi ffi fl .
We identify

∇p δ pxq " 2cµ « B 1 gpxqhpxq `gpxqB 1 hpxq B 2 gpxqhpxq `gpxqB 2 hpxq ff with gpxq " x 1 f 1 `x2 f 2 et hpxq " 1 }x} 2 .
Finally, the pressure p δ is given by

p δ pxq " 2cµ ˆx1 f 1 `x2 f 2 }x} 2 ˙`p 8 " 2cµ x ¨F }x} 2 `p8 ,
where p 8 is the value of the pressure at infinity. In what follows, we choose p 8 " 0. In order to set the constant c, we fix a function ϕ P C 8 0 pR 2 , R 2 q and assume that ϕ is radial (to ease calculations): ϕpxq " ϕp}x}q. For F " t r1, 0s:

x´µ△u δ `∇p δ , ϕy " żż R 2 ´´µ△u 1 pxqϕ 1 pxq ´µ△u 2 pxqϕ 2 pxq `∇p δ pxq ¨ϕpxq ¯dx " ´żż R 2 ´µu 1 pxq△ϕ 1 pxq `µu 2 pxq△ϕ 2 pxq ¯dx ´żż R 2 p δ pxq ´B1 ϕ 1 pxq `B2 ϕ 2 pxq ¯dx " ´cµ ˆżż R 2 ´ln }x}△ϕ 1 pxqdx `żż R 2 x 2 1 }x} 2 △ϕ 1 pxqdx `żż R 2 x 1 x 2 }x} 2 △ϕ 2 pxqdx `żż R 2 2x 1 }x} 2 pB 1 ϕ 1 pxq `B2 ϕ 2 pxqqdx " ´cµ ˆI1 `I2 `I3 `I4
Ẇe recall the following results:

ż 2π 0 cospθq sinpθqdθ " 0 and ż 2π 0 cos 2 pθqdθ " π.
' Calculation of I 1 :

I 1 " żż R 2 ´ln }x}△ϕ 1 pxqdx " ´ż 2π 0 ż `8 0 r ln r ˆB2 r,r ϕ 1 prq `1 r B r ϕ 1 prq ˙drdθ " ´ż 2π 0 ˜"r ln rB r ϕ 1 prq  `8 0 `ż `8 0 B r ϕ 1 prq pln r ´ln r ´1q dr ¸dθ " ´ż 2π 0 ż `8 0 ´Br ϕ 1 prqdrdθ " ´2πϕ 1 p0q.
' Calculation of I 2 :

I 2 " żż R 2 x 2 1 }x} 2 △ϕ 1 pxqdx " ż 2π 0 ż `8 0 r cos 2 pθq ˆB2 r ϕ 1 prq `1 r B r ϕ 1 prq ˙drdθ " ż 2π 0 cos 2 pθqdθ ż `8 0 ˆrB 2 r ϕ 1 prq `Br ϕ 1 prq ˙dr " π ˜"rB r ϕ 1 prq  `8 0 ´ż `8 0 B r ϕ 1 prqdr `ż `8 0 B r ϕ 1 prqdr " 0.
1.4. Elliptic problems with a singular right-hand side 71 ' Calculation of I 3 :

I 3 " żż R 2 x 1 x 2 }x} 2 △ϕ 2 pxqdx " ż 2π 0 ż `8 0 r cospθq sinpθq ˆB2 r ϕ 2 prq `1 r B r ϕ 2 prq ˙drdθ " ż 2π 0 cospθq sinpθqdθ ż `8 0 ˆrB 2 r ϕ 2 prq `Br ϕ 2 prq ˙dr " 0.
' Calculation of I 4 :

I 4 " żż R 2 2x 1 }x} 2 ˆB1 ϕ 1 pxq `B2 ϕ 2 pxq ˙dx " ż 2π 0 ż `8 0 2 cospθq ˆcospθqB r ϕ 1 prq `sinpθqB r ϕ 2 prq ˙drdθ " 2 ż 2π 0 cos 2 pθqdθ ż `8 0 B r ϕ 1 prqdr `2 ż 2π 0 cospθq sinpθqdθ ż `8 0 B r ϕ 2 prqdr " ´2πϕ 1 p0q.
Finally, we get x´µ△u δ `∇p δ , ϕy " 4πcµϕ 1 p0q, and we want x´µ△u δ `∇p δ , ϕy " xδ 0 , ϕ 1 y " ϕ 1 p0q.

We conclude that

c " 1 4πµ .

The same calculations with F " t r0, 1s give the same constant c. By linearity, it holds for any F in R 2 , which ends the proof.

Proposition 7. In 3-dimension, the "Stokeslet" is defined by

u δ pxq " 1 8πµ ˆI3 }x} `x ¨tx }x} 3 ˙¨F P R 3 , " 1 8πµ » - - - - - - - - f 1 }x} `x2 1 f 1 `x1 x 2 f 2 `x1 x 3 f 3 }x} 3 f 2 }x} `x1 x 2 f 1 `x2 2 f 2 `x2 x 3 f 3 }x} 3 f 3 }x} `x1 x 3 f 1 `x2 x 3 f 2 `x2 3 f 3 }x} 3 fi ffi ffi ffi ffi ffi ffi ffi fl P R 3 , p δ pxq " x ¨F 4π}x} 3 " x 1 f 1 `x2 f 2 `x3 f 3 4π}x} 3 P R, (1.4.4)
where F " t rf 1 , f 2 , f 3 s and x " t rx 1 , x 2 , x 3 s.

Proof. The proof is the same as in 2-dimension.

Regularity of the solution pu, pq

Let us come back on the focus of Problem (1.3.14). The main difficulty in the study of this kind of problem is the singularity of the source term (and therefore of the solution). In 2-dimension, Problem (1.3.14) has a unique weak solution pu, pq P W 1,q pΩq 2 ˆLq 0 pΩq for all q P r1, 2r. Actually, the 2d Stokeslet pu δ , p δ q satisfies Problem (1.4.1), so that the Stokeslet pu δ p¨´x 0 q, p δ p¨´x 0 qq contains the singular part of pu, pq, the solution of Problem (1.3.14). The solution pu, pq can be built by using a suitable lift procedure which consists in adding to pu δ , p δ q a corrector term pw, πq P H 1 pΩq 2 ˆL2 0 pΩq which satisfies the following problem:

$ & % ´△w `∇π " 0 in Ω, divpwq " 0 in Ω, w " ´uδ p¨´x 0 q on BΩ.
Then, the solution is given by: upxq " u δ pxq `wpxq "

1 4π ˆ´ln }x}I 2 `x ¨tx }x} 2 ˙F `wpxq, ppxq " p δ pxq `πpxq " x ¨F 2π}x} 2 `πpxq.
Moreover, it is easy to show that u δ R H 1 0 pΩq 2 and p δ R L 2 pΩq. Actually, we can specify how the quantity |u δ | 1,q,Ω goes to infinity when q goes to 2, with q ă 2 (which is noted q Ñ ă 2). By (1.4.2), estimating |u δ | 1,q,Ω when q Ñ ă 2 reduces to estimate |u δ | 1,q,B , where B denotes the ball Bpx 0 , 1q: we can easily show that there exists C ą 0 depending only on F such that @1 ď q ă `8, u δ P L q pΩq and |∇u δ | ď C }x} , and so, using polar coordinates, we get for q ă 2,

|u δ | q 1,q,Ω " ż B |∇u δ pxq| q dx ď ż B C q }x} q dx " C q ż 1 0 ż 2π 0 1 r q´1 dθdr " 2πC q 1 2 ´q .
Finally, there exists C ą 0 independent of q such that, for 1 ď q ă 2,

|u δ | 1,q,Ω ď C ? 2 ´q . (1.4.5)
In the same way, we can easily show that there exists C ą 0 independent of q such that, for 1 ď q ă 2,

|p δ | 0,q,Ω ď C ? 2 ´q . (1.4.6)
In 3-dimension, the punctual force source term is more singular, and the solution pu, pq belongs to W 1,q pΩq 2 ˆLq 0 pΩq for all q P r1, 3{2r. The construction of the solution pu, pq is the same as in 2d, and we have: there exists C ą 0 independent of q such that, for 1 ď q ă 3{2,

|u δ | 1,q,Ω ď C p3 ´2qq 2{3 and |p δ | 0,q,Ω ď C p3 ´2qq 2{3 .
Let us now do the same developments for the Poisson problem with a Dirac mass right-hand side, which is a scalar version of Problem (1.3.14).

Stokeslet in a confined domain

Even if Problem (1.3.14) is singular, each finite element space V h we consider satisfies A numerical solution of the problem 2d is illustrated in Figure 1. [START_REF] Boyer | Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model[END_REF], where Ω is the unit square, x 0 " p0.5, 0.5q and F " t r1, 1s. Notice that there are recirculations due to the boundedness of the domain.

V h Ă C p Ωq,

The Poisson problem with a Dirac mass right-hand side

In this section, we consider the solution u of the following Poisson problem:

" ´△u " δ x 0 in Ω, u " 0 on BΩ, (1.4.7)
where Ω is a bounded open smooth subdomain of R d , d " 2 or 3, and x 0 P Ω. As said at the beginning of this section, we focus also on this problem because it is a scalar version of Problem (1.3.14), and so a simplier problem to lead a preliminary study (see Chapter 2).

As it has been done in the case of the Stokes problem, we lead the study in 2-dimension and the results will be also given in 3-dimension.

The Green's function

Also for the sake of simplicity, the location of the punctual force is the origin. Let us consider the Poisson problem on the whole domain

´△u δ " δ 0 P R 2 . (1.4.8)
The solution of this problem is not unique: if u is a solution and v satisfies △v " 0, then u `v is another solution. We conclude that ´△u δ " δ 0 involves a " ´1 2π , and so u δ pxq " ´1 2π ln }x}.

Remark 2. Actually, we have shown that there exists an unique radial solution of Problem (1.4.8) up to an additive constant. The Green's function is the one for which this constant is zero.

Proposition 9. In 3-dimension, the Green's function is given by

u δ pxq " 1 4π 1 }x} .
Proof. The proof is the same as in 2-dimension.

Regularity of the solution u

In this subsection, we focus on the singularity of the solution, which is the main difficulty in the study of this kind of problems. In 2-dimension, Problem (1.4.7) has a unique weak solution u P W 1,q 0 pΩq for all q P r1, 2r (see for instance [START_REF] Araya | A posteriori error estimates for elliptic problems with Dirac delta source terms[END_REF]). In fact, the Green's function satisfies ´△u δ " δ 0 , so that u δ p¨´x 0 q contains the singular part of u. As it is done in [START_REF] Araya | A posteriori error estimates for elliptic problems with Dirac delta source terms[END_REF], the solution u can be built by adding to u δ p¨´x 0 q a corrector term ω P H 1 pΩq, solution of the Laplace Problem " ´△ω " 0 in Ω, ω " ´uδ p¨´x 0 q on BΩ.

(1.4.9)

Then, the solution of Problem (1.4.7) is given by upxq " u δ px ´x0 q `ωpxq " ´1 2π logp|x ´x0 |q `ωpxq.

It is easy to verify that u R H 1 0 pΩq. Actually, we can specify how the quantity }u} 1,q,Ω goes to infinity when q goes to 2, with q ă 2. By the foregoing, if we write u " u δ `ω, since ω P H 1 pΩq, estimating }u} 1,q,Ω when q converges to 2 from below is reduced to estimate }u δ } 1,q,B , where B " Bp0, 1q: u δ P L q pΩq for all 1 ď q ă 8, and using polar coordinates, we get, for q ă 2,

|u δ | q 1,q,B " ż B |∇u δ pxq| q dx " ż 1 0 ż 2π 0 ˆ1 2πr ˙q rdθdr " p2πq 1´q ż 1 0 r 1´q dr " p2πq 1´q 2 ´q .
Finally, when q Ñ ă 2,

}u} 1,q,Ω " 1 ? 2π 1 ? 2 ´q .
(1.4.10)

Illustration of the solution u

In this case again, even if Problem (1.4.7) is singular, the finite element solution is welldefined. The numerical analysis of this problem is led in Chapter 2. Figure 1.20 illustrates the Green's function in a confined domain, with Ω the unit square and x 0 its center, x 0 " p0.5, 0.5q.

The study of the regularity of the solution of the Poisson problem (1.4.7) ends this chapter. The next chapter is devoted to the analysis of the finite element method applied to solve the Poisson problem with a Dirac mass right-hand side. In Chapter 3, we lead the study in the case of the Stokes problem with a punctual force in source term. In the last chapter, we detail a new numerical method to solve these problems and present numerical results got for the mucociliary transport.

CHAPTER 2

LOCAL ERROR ESTIMATES OF THE FINITE ELEMENT METHOD FOR THE POISSON PROBLEM WITH A DIRAC RIGHT-HAND SIDE

The solutions of elliptic problems with a Dirac measure right-hand side are not H 1 and therefore the convergence of the finite element solutions is suboptimal. The use of graded meshes is standard remedy to recover quasi-optimality, namely optimality up to a log-factor, for low order finite elements in the L 2 -norm. Optimal (or quasi-optimal for the lowest order case) convergence for Lagrange finite elements has been shown, in the L 2 -norm, on a subdomain which excludes the singularity. Here, on such subdomains, we show a quasi-optimal convergence in the H s -norm, for s ě 1, and, in the particular case of Lagrange finite elements, an optimal convergence in H 1 -norm, on a family of quasiuniform meshes in dimension 2. The study of this problem is motivated by the use of the Dirac measure as a reduced model in physical problems, for which high accuracy of the finite element method at the singularity is not required. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.

Introduction

This chapter deals with the accuracy of the finite element method on the Poisson problem with a singular right-hand side. More precisely, let us consider the Dirichlet problem

pP δ q " ´△u " δ x 0 in Ω, u " 0 on BΩ,
where Ω Ă R 2 is a bounded open C 8 domain or a square, and δ x 0 denotes the Dirac measure concentrated at a point x 0 P Ω such that distpx 0 , BΩq ą 0.

Problems of this type occur in many applications from different areas, like in the mathematical modeling of electromagnetic fields [START_REF] Jackson | Classical electrodynamics[END_REF]. Dirac measures can also be found on the right-hand side of adjoint equations in optimal control of elliptic problems with state constraints [START_REF] Casas | Control of an elliptic problem with pointwise state constraints[END_REF]. As further examples where such measures play an important role, we mention controllability for elliptic and parabolic equations [START_REF] Casas | Parabolic control problems in measure spaces with sparse solutions[END_REF][START_REF] Casas | Spike controls for elliptic and parabolic PDEs[END_REF][START_REF] Leykekhman | Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints[END_REF] and parameter identification problems with pointwise measurements [START_REF] Rannacher | A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements[END_REF].

Our interest in pP δ q is motivated by the modeling of the movement of a thin structure in a viscous fluid, such as cilia involved in the muco-ciliary transport in the lung [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF]. In the asymptotic of a zero diameter cilium with an infinite velocity, the cilium is modelled by a line Dirac of force in the source term. In order to make the computations easier, the line Dirac of force can be approximated by a sum of punctual Dirac forces distributed along the cilium [START_REF] Lacouture | A numerical method to solve the Stokes problem with a punctual force in source term[END_REF]. In this chapter, we address a scalar version of this problem: problem pP δ q.

In the regular case, namely the Poisson problem with a regular right-hand side, the finite element solution u h is well-defined and for u P H k`1 pΩq, we have, for all 0 ď s ď 1,

}u ´uh } s ď Ch k`1´s }u} k`1 , (2.1.1)
where k is the degree of the method [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] and h the mesh size. As can be noticed, the efficiency of the method critically depends on regularity of the solution. We aim at describing the impact of the loss of the regularity of the solution over the classical estimates Chapter 2. Local error estimates of the finite element method for the Poisson problem with a Dirac right-hand side due to irregular source term. In dimension 1, the solution u of Problem pP δ q belongs to H 1 pΩq, but it is not H 2 pΩq. In this case, the numerical solution u h and the exact solution u can be computed explicitly. If x 0 matches with a node of the discretization, u h " u.

Otherwise, this equality is true only on the complementary of the element which contains x 0 , and the convergence orders are 1/2 and 3/2 respectively in H 1 -norm and L 2 -norm.

In dimension 2, Problem pP δ q has no H 1 pΩq-solution, and so, although the finite element solution can be defined, the H 1 pΩq-error has no sense and the L 2 pΩq-error estimates cannot be obtained by the Aubin-Nitsche method without modification.

Let us review the literature about error estimates for problem pP δ q, starting with discretizations on quasi-uniform meshes. Babuska [START_REF] Babuška | Error-bounds for finite element method[END_REF] showed an L 2 pΩq-convergence of order h 1´η , η ą 0, for a two-dimensional smooth domain. Scott proved in [START_REF] Scott | Finite element convergence for singular data[END_REF] an a priori error estimates of order 2 ´d 2 , where the dimension d is 2 or 3. The same result has been proved by Casas [START_REF] Casas | L 2 estimates for the finite element method for the Dirichlet problem with singular data[END_REF] for general Borel measures on the right-hand side.

To the best of our knowledge, in order to improve the convergence order, Eriksson [START_REF] Eriksson | Improved accuracy by adapted mesh-refinements in the finite element method[END_REF] was the first who studied the influence of locally refined meshes near x 0 . Using results from [START_REF] Schatz | Maximum norm estimates in the finite element method on plane polygonal domains[END_REF], he proved convergence of order k and k `1 in the W 1,1 pΩq-norm and the L 1 pΩq-norm respectively, for approximations with a P k -finite element method. Recently, by Apel and co-authors [2], an L 2 pΩq-error estimate of order h 2 | ln h| 3{2 has been proved in dimension 2, using graded meshes. Optimal convergence rates with graded meshes were also recovered by D'Angelo [START_REF] Angelo | Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces : applications to one-and three-dimensional coupled problems[END_REF] using weighted Sobolev spaces. A posteriori error estimates in weighted spaces have been established by Agnelli and co-authors [START_REF] Agnelli | A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces[END_REF].

These theoretical a priori results are based upon graded meshes, which increase the complexity of the meshing and the computational cost, even if the mesh is refined only locally, especially when the right-hand side includes several Dirac measures, that can be static or moving. Therefore Eriksson [START_REF] Eriksson | Finite element methods of optimal order for problems with singular data[END_REF] developed a numerical method to solve the problem and recover the optimal convergence rate: the numerical solution is searched in the form u 0 `wh where u 0 contains the singularity of the solution and w h is the numerical solution of a smooth problem. This method is presented in the case of the Poisson and the Stokes problems in [START_REF] Lacouture | A numerical method to solve the Stokes problem with a punctual force in source term[END_REF].

However, in applications, the Dirac measure at x 0 is often a model reduction approach, and a high accuracy at x 0 of the finite element method is not necessary. Thus, it is interesting to study the error on a fixed subdomain which excludes the singularity. Recently, Köppl and Wohlmuth have shown in [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF] optimal convergence in L 2 -norm for the Lagrange finite elements (the result is quasi-optimal for the P 1 -element). In this chapter, we consider the problem in dimension 2, and we show :

1. Quasi-optimal convergence in H s -norm, for s ě 1. This result applies to a wide class of finite-element methods and beyond, including Lagrange and Hermite finite elements and wavelets. The L 2 -error estimates established in [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF] are not used and the proof is based on different arguments.

2. Optimal convergence in H 1 -norm for the Lagrange finite elements. This result is obtained by direct use of the optimal L 2 -norm convergence result in [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF].

3. Optimal convergence in H 1 -norm in the particular case of the P 1 -Lagrange finite element using different arguments than those used for the previous results.

These results imply that graded meshes are not required to recover optimality far from the singularity and that there are no pollution effects. In addition, by linearity of Problem pP δ q, the result holds in the case of several Dirac masses. The chapter is organized as follows. We begin with the 1-dimensional case, Section 2.2. Our main results are presented in Section 2.3 after recalling the Nitsche and Schatz Theorem, which is an important tool for the proof presented in Section 2.4. In Section 2.5 another argument is presented to obtain an optimal estimate in the particular case of the P 1 -finite elements. We illustrate in Section 2.6 our theoretical results by numerical simulations and, in Section 2.7, we discuss the generalization of our approach to the three-dimensional case.

Preliminaries: one-dimensional case

Before getting to the heart of the matter, let us consider the problem in dimension 1. We consider I "sa, br and suppose that the Dirac mass δ is supported by x 0 Psa, br. In dimension 1, the problem pP δ q can be written as pP 1 δ q " ´u2 " δ x 0 in sa, br, upaq " upbq " 0.

It is easy to show that the solution u of the Poisson problem pP 1 δ q (drawn on In dimension 1 (and only in dimension 1), the solution u belongs to H 1 0 pIq (actually, it belongs to H 3{2´η pIq for all η ą 0). In this section, the study is limited to the case of the Lagrange finite elements P 1 but the results can be easily extended to P k -finite elements. Moreover, it is possible to write an explicit form of the numerical solution u h . Theorem 2. Let T h be a discretization of the interval I "sa, br, the numerical solution u h satisfies one of the two following assertions :

' if x 0 is a point of the discretization T h (mesh 1 on Figure 2.1), u h " u.
' if x 0 is not a point of the discretization T h (mesh 2 on Figure 2.1), by denoting x i and x i`1 the two consecutive points of the discretization T h such as x i ă x 0 ă x i`1 , the numerical solution u h is equal to the exact solution u on the closed sub-domain ra, x i s Ť rx i`1 , bs, it is continue on I and affine on rx i , x i`1 s (see Figure 2.1). Actually, the numerical solution u h is equal on I to the interpolation I h u. Moreover, the convergence can be specified : Remark 3. In dimension 1, the discretization T h does not need to be uniform or quasiuniform. Estimates (2.2.2) are true for h " x i`1 ´xi , the length of the segment which contains the point x 0 .

}u ´uh }
Proof. If x 0 is a point of the discretization T h , u belongs to the approximation space V h , and a direct consequence of the Cea lemma is u h " u. Otherwise, let us look at the discrete right-hand side corresponding to the Dirac measure δ x 0 : it is of the form

i í `1 » - - - - - - - - - 0 . . . α β . . . 0 fi ffi ffi ffi ffi ffi ffi ffi ffi fl " » - - - - - - - - - 0 . . . α 0 . . . 0 fi ffi ffi ffi ffi ffi ffi ffi ffi fl `» - - - - - - - - - 0 . . . 0 β . . . 0 fi ffi ffi ffi ffi ffi ffi ffi ffi fl
, where α " v i px 0 q and β " v i`1 px 0 q, with v i and v i`1 the test functions corresponding to the points x i and x i`1 of the mesh. By linearity of problem pP 1 δ q, the numerical solution u h is given by

u h " u h α `uh β ,
where u h α and u h β are respectively the numerical solutions of problems pP α q " ´u2 α " αδ x i in sa, br, u α paq " u α pbq " 0, and pP β q " ´u2 β " βδ x i`1 in sa, br, u β paq " u β pbq " 0.

Each of these problems satisfies the first item of Theorem 2, and thus, by (2.2.1),

u h α pxq " $ ' ' & ' ' % α b ´xi b ´a px ´aq if x P ra, x i s, α x i ´a b ´a pb ´xq if x P rx i , bs, u h β pxq " $ ' ' & ' ' % β b ´xi`1 b ´a px ´aq if x P ra, x i`1 s, β x i`1 ´a b ´a pb ´xq if x P rx i`1 , bs, with α " x i`1 ´x0 x i`1 ´xi and β " x 0 ´xi x i`1 ´xi .
Finally, since the function u h " u h α `uh β is continuous and affine on each of the segments ra, x i s, rx i , x i`1 s and rx i`1 , bs, it remains only to verify that the function u h satisfies u h paq " u h pbq " 0, u h px i q " upx i q and u h px i`1 q " upx i`1 q. Now that the analytic expressions of u and u h are known, error Estimates (2.2.2) can be deduced from a direct calculation of the error on the segment rx i , x i`1 s. Remark 4. Since in one dimension the solution u belongs to H 1 pIq, the finite element method is well-defined, but the solution does not belong to H 2 pIq, and thus the classical error Estimates (2.1.1) are no longer valid: the order of convergence is 1/2 in H 1 -norm and 3/2 in L 2 -norm against respectively 1 and 2 in the regular case.

We have proved with Theorem 2 that in dimension 1 the numerical solution is equal to the exact solution outside the segment of the mesh which contains the singularity. Moreover, if the point x 0 matches with a point of the discretization, the numerical solution is actually the exact solution. Thus, if the mesh can be chosen conformed to the singularity, the error of the finite element method is zero. The following focuses on the local error far from the Dirac mass in dimension 2. As it will be seen, the 1-dimension arguments cannot be used anymore.

Main results

In this section, we define all the notations used in this chapter, recall an important tool for the proof, the Nitsche and Schatz Theorem, and formulate our main results. For a domain D, we denote by } ¨}s,q,D (respectively | ¨|s,q,D ) the norm (respectively the semi-norm) of the Sobolev space W s,q pDq, while } ¨}s,D (respectively | ¨|s,D ) stand for the norm (respectively the semi-norm) of the Sobolev space H s pDq.

For the numerical solution, let us introduce a family of quasi-uniform simplicial triangulations T h of Ω and a finite element space V k h Ă H 1 0 pΩq, where k is the order of approximation. To ensure that the numerical solution is well-defined, the space V k h is assumed to contain only continuous functions. The finite element solution u h P V k h of problem pP δ q is defined by ż

Ω ∇u h ¨∇v h " v h px 0 q, @v h P V k h . (2.3.1)
We also evaluate the H s -norm of the error on a subdomain of Ω which does not contain the singularity, for s ě 2, and, whenever we do so, we of course assume the finite elements are H s -conform. We fix two subdomains of Ω, named Ω 0 and Ω 1 , such that Ω 0 ĂĂ Ω 1 ĂĂ Ω and x 0 R Ω 1 (see Figure 2.2). We consider a mesh which satisfies the following condition:

Assumption 1. For some h 0 , we have for all 0 ă h ď h 0 (see Figure 2.2),

Ω m 0 Ş Ω c 1 " H, where Ω m 0 " ď T PT h T Ş Ω 0 ‰H T,
and Ω c 1 is the complement of Ω 1 in Ω.

Nitsche and Schatz Theorem

Before stating the Nitsche and Schatz Theorem, let us introduce properties satisfied by the finite element spaces V k h .

Assumption 2. Given two fixed concentric spheres B 0 and B with B 0 ĂĂ B ĂĂ Ω, there exists an h 0 such that for all 0 ă h ď h 0 , we have for some R ě 1 and M ą 1:

A1 For any 0 ď s ď R and s ď ℓ ď M, for each v P H ℓ pBq, there exists η P V k h such that }v ´η} s,B ď Ch ℓ´s }v} ℓ,B .

Moreover, if v P H 1 0 pB 0 q then η can be chosen to satisfy η P H 1 0 pBq. A2 Let ϕ P C 8 0 pB 0 q and v h P V k h , then there exists

η P V k h Ş H 1 0 pBq such that }ϕv h ´η} 1,B ď Cpϕ, B, B 0 qh}v h } 1,B .
A3 For each h ď h 0 there exists a domain B h with B 0 ĂĂ B h ĂĂ B such that if 0 ď s ď ℓ ď R then for all v h P V k h we have

}v h } ℓ,B h ď Ch s´ℓ }v h } s,B h .
We now state the following theorem, a key tool in the forthcoming proof of Theorem 3.

Theorem (Nitsche and Schatz [START_REF] Nitsche | Interior estimates for Ritz-Galerkin methods[END_REF]). Consider Ω 0 ĂĂ Ω 1 ĂĂ Ω and let V k h satisfy Assumption 2. Let v P H ℓ pΩ 1 q, let v h P V k h and let t be a nonnegative integer, arbitrary but fixed. Let us suppose that v ´vh satisfies ż Ω ∇pv ´vh q ¨∇η " 0, @η P V k h Ş H 1 0 pΩ 1 q.

Then there exists h 1 such that if h ď h 1 we have (i) for s " 0, 1 and 1 ď ℓ ď M,

}v ´vh } s,Ω 0 ď C `hℓ´s }v} ℓ,Ω 1 `}v ´vh } ´t,Ω 1 ˘, (ii) for 2 ď s ď ℓ ď M and s ď k ă R, }v ´vh } s,Ω 0 ď C `hℓ´s }v} ℓ,Ω 1 `h1´s }v ´vh } ´t,Ω 1 ˘.
For our study, we actually need a more general version of the assumptions on the approximation space V k h : Assumption 3. Given B Ă Ω and q 1 ě 2, there exists an h 0 such that for all 0 ă h ď h 0 , we have for some R ě 1 and M ą 1: r A1 For any 0 ď s ď R and s ď ℓ ď M, for each v P H ℓ pBq, there exists η P V k h such that, for any finite element T Ă B, |v ´η| s,q 1 ,T ď Ch dp1{q 1 ´1{2q h ℓ´s |v| ℓ,2,T .

Chapter 2. Local error estimates of the finite element method for the

Poisson problem with a Dirac right-hand side r A3 For 0 ď s ď ℓ ď R, for all v h P V k h , for any finite element T in the family T h , we have }v h } ℓ,q 1 ,T ď Ch 2p1{q 1 ´1{2q h s´ℓ }v h } s,2,T .

Assumptions r A1 and r A3 are generalizations of Assumptions A1 and A3. They are quite standard and satisfied by a wide variety of approximation spaces, including all finite element spaces defined on quasi-uniform meshes [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]. The parameters R and M play respectively the role of the regularity and order of approximation of the approximation space V k h . For example, in the case of P 1 -finite elements, we have R " 3{2 ´ε and M " 2. Assumption A2 is less common but also satisfied by a wide class of approximation spaces. Actually, for Lagrange and Hermite finite elements, a stronger property than Assumption A2 can be shown as a consequence of Assumptions A1 and A3: Lemma 1. Let us consider Lagrange finite elements, or Hermite finite elements if s ą 1, we still denote by V k h the approximation space of degre k. Given two concentric spheres B 0 and B with B 0 ĂĂ B ĂĂ Ω, and let us fix ϕ P C 8 0 pB 0 q. For any 0 ď s ď ℓ ď k and any v h P V k h , there exists η P V k h Ş H 1 0 pBq such as

}ϕv h ´η} s,B ď Cpϕqh ℓ´s`1 }v h } ℓ,B .
Applied for s " ℓ " 1, Lemma 1 gives assumption A2. The proof is inspired from the ones presented in a more general case in [START_REF] Bertoluzza | The discrete commutator property of approximation spaces[END_REF].

Proof. Let 0 ď s ď k. Since ϕ P C 8 0 pBq, by Assumption A1, there exists

η P V k h Ş H 1 0 pBq such that }ϕv h ´η} s,B ď Ch k`1´s }ϕv h } k`1,B .

So we can write

}ϕv h ´η} 2 s,B ď Ch 2pk`1´sq ÿ T PT h }ϕv h } 2 k`1,T .
In order to bound the quantity }ϕv h } k`1,T , we need the following result:

Proposition 10 (Ciarlet, page 192 [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]). Let G be a subset of R d and an integer k ě 1, let a P W k,8 pGq and b P H k pGq. Then ab P H k pGq and

}ab} k,G ď C p|a| 0,8,G |b| k,G `}a} k,8,G }b} k´1,G q .
Applied for G " T P T h , a " ϕ and b " v h , Proposition 10 gives

}ϕv h } k`1,T ď C p|ϕ| 0,8,T |v h | k`1,T `}ϕ} k,8,T }v h } k,T q .
Since v h P R k rXs, |v h | k`1,T " 0, and so

}ϕv h } k`1,T ď C}ϕ} k,8,T }v h } k,T .
Thus, we get: there exists η P V k h Ş H 1 0 pBq such that

}ϕv h ´η} s,B ď Cpϕqh k`1´s }v h } k,B .
To end the proof, we use Assumption A3: for any 0 ď ℓ ď k,

}v h } k,B ď Ch ℓ´k }v h } ℓ,B .
Finally, there exists η P V k h Ş H 1 0 pBq such that }ϕv h ´η} s,B ď Cpϕqh ℓ´s`1 }v h } ℓ,B .

Statement of our main results

Our main results are Theorems 3, 4 and 5. The rest of the chapter is mostly concerned by the proof and the illustration of these results.

Theorem 3. Let Ω 0 ĂĂ Ω 1 ĂĂ Ω satisfy Assumption 1, 1 ď s ď k.
Let u be the solution of problem pP δ q and u h its Galerkin projection onto V k h , satisfying (2.3.1). Under Assumptions 2 and 3, there exists h 1 such that if 0 ă h ď h 1 , we have,

}u ´uh } 1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k a | ln h|. (2.3.2)
In addition, for s ě 2, if the finite elements are supposed H k -conforming, we have

}u ´uh } s,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k`1´s a | ln h|. (2.3.3)
Remark 5. The main tool in proving Theorem 3 is the Nitsche and Schatz Theorem, and the result holds for all the spaces verifying Assumptions 2 and 3. The class of such spaces includes spaces beyond finite elements, including, for instance, wavelets.

Section 2.4 is dedicated to the proof of Theorem 3.

In the particular case of Lagrange finite elements, Köppl and Wohlmuth [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF] showed, in the L 2 -norm of a subdomain which does not contain x 0 , quasi-optimality for the lowest order case, and optimal a priori estimates for higher order. The proof is based on Wahlbintype arguments, which are similar to the Nitsche and Schatz Theorem (see [START_REF] Wahlbin | Local behavior in finite element methods[END_REF][START_REF] Wahlbin | Superconvergence in Galerkin finite element methods[END_REF]), and different arguments from the ones presented in this chapter, like the use of an operator of Scott and Zhang type [START_REF] Scott | Finite element interpolation of nonsmooth functions satisfying boundary conditions[END_REF]. Using this result it is possible to prove quite easily optimal convergence in H 1 -norm for Lagrange finite elements. This result reads as follows:

Theorem 4. Consider a domain Ω 2 such that Ω 0 ĂĂ Ω 1 ĂĂ Ω 2 ĂĂ Ω, x 0 R Ω 2 ,
and satisfying Assumption 1. Let u be the solution of problem pP δ q and u h its Galerkin projection onto the space of Lagrange finite elements of order k `1. There exists h 1 such that if 0 ă h ď h 1 , we have

}u ´uh } 1,Ω 0 ď CpΩ 1 , Ω 2 , Ωqh k .
Chapter 2. Local error estimates of the finite element method for the Poisson problem with a Dirac right-hand side Remark 6. This result is optimal and thus slightly stronger than Inequality (2.3.2), but it is limited to Lagrange finite elements and to the H 1 -norm, due to the use of an operator of Scott-Zhang type. Theorem 3 is more general: it holds for a wide class of finite elements and it allows to estimate the error in H s -norm, for any s ě 1.

Proof of Theorem 4. In the particular case of Lagrange finite elements, Köppl and Wohlmuth proved in [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF] the following convergence in the L 2 -norm of a subdomain which does not contain x 0 :

}u ´uh } 0,Ω 1 ď CpΩ 1 , Ω 2 , Ωq " h 2 | lnphq| if k " 1, h k`1 if k ą 1. (2.3.4)
Let us apply the Nitsche and Schatz Theorem on Ω 0 and Ω 1 for l " k `1 and t " 0,

}u ´uh } 1,Ω 0 ď C `hk }u} 2,Ω 1 `}u ´uh } 0,Ω 1 ˘.
Using (2.3.4), we get }u ´uh } 1,Ω 0 ď Ch k .

For the particular P 1 -Lagrange finite elements, we prove the optimal convergence in H 1 -norm using completely different arguments. This proof involves a technical assumption on the mesh, namely Assumption 4 in Section 2.5.2: the distance of the Dirac mass to the edges of the mesh triangles is assumed to be at least of the same order as the mesh size h. The result reads as follows:

Theorem 5. Let Ω 0 ĂĂ Ω 1 ĂĂ Ω satisfy Assumption 1 and consider a mesh such that there exists a domain B ε satisfying Assumption 4 with ε of the same order as the mesh size. The P 1 -finite element method converges with order 1 for the H 1 pΩ 0 q-norm. More precisely:

}u ´uh } 1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh.
The proof of this result is detailed in Section 2.5.

Proof of Theorem 3

This section is devoted to the proof of Theorem 3. We first show a weak version of the Aubin-Nitsche duality lemma (Lemma 2) and establish a discrete inf-sup condition (Lemma 3). Then, we use these results to prove Theorem 3.

Aubin-Nitsche duality lemma with a singular right-hand side

The proof of Theorem 3 is based on Nitsche and Schatz Theorem. In order to estimate the quantity }u ´uh } ´t,Ω 1 , we first show a weak version of Aubin-Nitsche Lemma, in the case of the Poisson problem with a singular right-hand side.

Lemma 2. Let f P W ´1,q pΩq " pW 1,q 1 0 pΩqq 1 , 1 ă q ă 2, and v P W 1,q 0 pΩq be the unique solution of " ´△v " f in Ω, v " 0 on BΩ.

Let v h P V k h be the Galerkin projection of v. For finite elements of order k, letting e " v ´vh , we have for all 0 ď t ď k ´1, }e} ´t,Ω ď Ch t`1 h 2p1{q 1 ´1{2q |e| 1,q,Ω .

(2.4.1)

Proof. We aim at estimating, for t ě 0, the H ´t-norm of the error e:

}e} ´t,Ω " sup φPC 8 0 pΩq | ş Ω eφ| }φ} t,Ω . (2.4.2)
The error e P W 1,q 0 pΩq satisfies ż Ω ∇e ¨∇η " 0, @η P V k h .

Let be φ P C 8 0 pΩq and let w φ P H t`2 be the solution of " ´△w φ " φ in Ω, w φ " 0 on BΩ.

In dimension 2, by the Sobolev injections established in [START_REF] Brézis | Analyse fonctionnelle : théorie et applications[END_REF], H t`2 pΩq Ă W 1,q 1 pΩq for all q 1 in r2, `8r. Thus, for any

w h P V k h , ˇˇˇż Ω eφ ˇˇˇ" ˇˇˇż Ω e△w φ ˇˇˇ" ˇˇˇż Ω ∇e ¨∇w φ ˇˇˇ" ˇˇˇż Ω ∇e ¨∇pw φ ´wh q ˇˇˇď |w φ ´wh | 1,q 1 ,Ω |e| 1,q,Ω .
We have to estimate |w φ ´wh | 1,q 1 ,Ω . It holds

|w φ ´wh | q 1 1,q 1 ,Ω " ÿ T |w φ ´wh | q 1 1,q 1 ,T .
For all 0 ď t ď k ´1 and for all element T in T h , thanks to Assumption r A1 applied for s " 1, ℓ " t `2, there exists w h P V k h such as

|w φ ´wh | 1,q 1 ,T ď Ch 2p1{q 1 ´1{2q h t`1 |w φ | t`2,2,T . (2.4.3)
We number the triangles of the mesh tT i , i " 1, ¨¨¨, Nu and we set a " pa i q i and b " pb i q i , where a i " |w φ ´wh | 1,q 1 ,T i and b i " |w φ | t`2,2,T i .

By (2.4.3), we have, for all i in rr1, Nss,

a i ď Ch 2p1{q 1 ´1{2q h t`1 b i .
Chapter 2. Local error estimates of the finite element method for the Poisson problem with a Dirac right-hand side

We recall the norm equivalence in R N for 0 ă r ă s, }x} ℓ s ď }x} ℓ r ď N 1{r´1{s }x} ℓ s .

Remark that here N " Ch ´2. As 2 ă q 1 , we have }b} ℓ q 1 ď }b} ℓ 2 . Then, we can write

|w φ ´wh | 1,q 1 ,Ω " }a} ℓ q 1 ď Ch t`1 h 2p1{q 1 ´1{2q }b} ℓ q 1 , ď Ch t`1 h 2p1{q 1 ´1{2q }b} ℓ 2 , ď Ch t`1 h 2p1{q 1 ´1{2q |w φ | t`2,2,Ω , ď Ch t`1 h 2p1{q 1 ´1{2q }φ} t,Ω .
Finally, using this estimate in (2.4.2), we obtain, for t ď k ´1, }e} ´t,Ω ď Ch t`1 h 2p1{q 1 ´1{2q |e| 1,q,Ω .

Corollary 1. For finite elements of order k, for any 0 ă ε ă 1,

}u ´uh } ´k`1,Ω ď Ch k h ´ε|u ´uh | 1,qε,Ω . (2.4.4)
where q ε Ps1, 2r is defined by

q ε " 2 1 `ε ˆand so q 1 ε " 2 1
´ε˙.

(2.4.5)

Proof. We apply Lemma 2 to estimate }u ´uh } ´t,Ω , for pq ε , q 1 ε q defined in (2.4.5). In Inequality (2.4.1): To achieve this, we need the following discrete inf-sup condition.

2 ˆ1 q 1 ε ´1 2 ˙" 2 ˆ1 ´ε 2 ´1 2 ˙" ´ε. ( 2
Lemma 3. For 0 ă ε ă 1, q ε and q 1 ε defined in (2.4.5), we have the discrete inf-sup condition ş Ω ∇u ¨∇v }u} 1,qε }v} 1,q 1 ε ě β ą 0 holds for β independent of q ε and q 1 ε (and thus independent of ε). It is a consequence of the duality of the two spaces W 1,qε 0 pΩq and W 1,q 1 ε 0 pΩq, see [START_REF] Jerison | The inhomogeneous Dirichlet problem in Lipschitz domains[END_REF].

inf u h PV k h sup v h PV k h ş Ω ∇u h ¨∇v h }u h } 1,qε,Ω }v h } 1,q 1 ε ,Ω ě Ch ε .
For v P W 1,q 1 ε 0 pΩq, let Π h v denote the H 1 0 -Galerkin projection of v onto V k h . This is well defined since W 1,q 1 ε 0
pΩq Ă H 1 0 pΩq. We apply Assumption r A3 to Π h v for ℓ " s " 1, and get

}Π h v} 1,q 1 ε ,Ω ď Ch ´2p1{2´1{q 1 ε q }Π h v} 1,2,Ω ď Ch ´2p1{2´1{q 1 ε q }v} 1,2,Ω ď Ch ´2p1{2´1{q 1 ε q }v} 1,q 1 ε ,Ω . Moreover, for any u h P V k h Ă W 1,qε pΩq, }u h } 1,qε,Ω ď C sup vPW 1,q 1 ε 0 ş Ω ∇u h ¨∇v }v} 1,q 1 ε ,Ω " C sup vPW 1,q 1 ε 0 ş Ω ∇u h ¨∇Π h v }v} 1,q 1 ε ,Ω ď Ch ´2p1{2´1{q 1 ε q sup vPW 1,q 1 ε 0 ş Ω ∇u h ¨∇Π h v }Π h v} 1,q 1 ε ,Ω ď Ch ´2p1{2´1{q 1 ε q sup v h PV k h ş Ω ∇u h ¨∇v h }v h } 1,q 1 ε ,Ω .
Finally, thanks to Poincaré inequality, and to Inequality (2.4.6),

inf u h PV k h sup v h PV k h ş Ω ∇u h ¨∇v h }u h } 1,qε,Ω }v h } 1,q 1 ε ,Ω ě Ch ε .
Then, we can estimate |u ´uh | 1,qε,Ω : Lemma 4. With q ε and q 1 ε defined in (2.4.5),

|u ´uh | 1,qε,Ω ď C h ´ε ? ε . (2.4.7) Proof. By Lemma 3, it exists v h P V k h , with }v h } 1,q 1 ε ,Ω " 1, such that h 2p1{2´1{q 1 ε q }u h } 1,qε,Ω ď C ż Ω ∇u h ¨∇v h " C ż Ω ∇u ¨∇v h ď C}u} 1,qε,Ω .
So we have

|u ´uh | 1,qε,Ω ď |u| 1,qε,Ω `|u h | 1,qε,Ω ď Ch ´2p1{2´1{q 1 ε q }u} 1,qε,Ω . ( 2

Proof of Theorem 3

We can now prove Theorem 3.

Proof. The function u is analytic on Ω 1 , therefore the quantity }u} k`1,Ω which ends the proof of Theorem 3.

Proof of Theorem 5

In the particular case of the P 1 -finite element method, different arguments from the ones presented in the general case permit us to recover optimality. The approach is as follows: we first regularize the right-hand side and prove that in our case the solution u of pP δ q and the solution of the regularized problem are the same on the complementary of a neighbourhood of the singularity (Theorem 6). The proof of Theorem 5 is based once again on the Nitsche and Schatz Theorem and the observation that the discrete righthand sides of problem pP δ q and the regularized problem are exactly the same, so that the numerical solutions are the same too (Lemma 6).

Direct problem and regularized problem

The results presented in this section are valid in any dimension d ě 1. However they are only applied in dimension 2 in Section 2.5.3 in order to prove Theorem 2. Let ε ą 0, and f ε be defined on Ω by

f ε " d σpS d´1 qε d 1 Bε , (2.5.1)
where B ε " Bpx 0 , εq and σpS d´1 q is the Lebesgue measure of the unit sphere in dimension d. The parameter ε is supposed to be small enough so that B ε ĂĂ Ω. The function f ε is a regularization of de the Dirac distribution δ x 0 . Let us consider the following problem:

pP ε q " ´△u ε " f ε in Ω, u ε " 0 on BΩ.
Since f ε P L 2 pΩq, it is possible to show that problem pP ε q has a unique variational solution

u ε in H 1 0 pΩq Ş H 2 pΩq [54]
. We show the following result: Theorem 6. The solution u of pP δ q and the solution u ε of pP ε q coincide on the closed set r

Ω " ΩzB ε , ie, u | r Ω " u ε | r Ω .
The proof is based on the following lemma.

Lemma 5. Let d P Nzt0u, ε ą 0, x P R d , v a function defined on R d , harmonic on Bpx, εq, and f P L 1 pR d q such that • f is radial and positive,

• supppf q Ă Bp0, εq, ε ą 0,

• ż R d f pxq dx " 1. Then, f ˚vpxq " ż R d
f pyqvpx ´yq dy " vpxq.

Proof. As supppf q Ă Bp0, εq, using spherical coordinates, we have:

f ˚vpxq " ż ε 0 ż S d´1
f prqvpx ´rωqr d´1 dω dr "

ż ε 0 r d´1 f prq ˆżS d´1 vpx ´rωqdω ˙dr.
Besides, v is harmonic on Bpx, εq, so that the mean value property gives, for 0 ă r ď ε, Proof. First, let us leave out boundary conditions and consider the following problem ´△v " f ε in D 1 pR d q.

(2.5.2)

With u δ defined in Chapter 1, Section 1.4.2.1, ´△u δ " δ 0 in D 1 pR d q, so we can build a function v satisfying (2.5.2) as:

vpxq " f ε ˚uδ pxq " ż R d f ε pyqu δ px ´yq dy " ż R d
f ε px 0 `yqu δ px ´x0 ´yq dy " ´f ε px 0 `¨q ˚uδ ¯px ´x0 q.

Moreover, for all x P ΩzB ε , u δ is harmonic on Bpx ´x0 , εq, and f ε p¨`x 0 q satisfies the assumptions of Lemma 5, so that vpxq " ´f ε px 0 `¨q ˚uδ ¯px ´x0 q " u δ px ´x0 q. We conclude that v and u δ p¨´x 0 q have the same trace on BΩ, and so v `ω, where ω is the solution of the Laplace Problem (1.4.9) (see Chapter 1, Section 1.4.2.2), is a solution of the problem pP ε q. By the uniqueness of the solution, we have u ε " v `ω. Finally, for all x P ΩzB ε , u ε pxq " upxq. Since these functions are continuous on r Ω " ΩzB ε , this equality is true on the closure of r Ω, which ends the proof of Theorem 6.

Remark 7. Theorem 6 holds for any radial nonnegative function f P L 1 pR d q Ş L 2 pR d q such that supppf q Ă Bp0, εq and ż

R d f pxq dx " 1, taking f ε " f p¨´x 0 q. It is a direct consequence of Lemma 5.
Remark 8. The result is true in dimension 1, taking

f ε " 1 2ε 1 Iε ,
where I ε " rx 0 ´ε, x 0 `εs Ăsa, br" I. In this case, we can easily write down the solutions u and u ε explicitly,

upxq " $ ' ' & ' ' % b ´x0 b ´a x ´ab ´x0 b ´a if x P

Discretizations of the right-hand side

At this point, we introduce a technical assumption on B ε and the mesh. Remark 9. The parameter ε will be chosen to be h{10, so it remains to fix a "good" triangle T 0 and to build the mesh accordingly, so that Assumption 4 is satisfied. Notice that it is always possible to locally modify any given mesh so that it satisfies this assumption. Lemma 6. Under Assumption 4, u ε h " u h , where u h and u ε h are respectively the numerical solutions of problems pP δ q and pP ε q.

Proof. Let us write down explicitly the discretized right-hand side F ε h associated to the function f ε : for all node i and associated test function

v i P V 1 h , `F ε h ˘i " ż Ω 1 σpB ε q 1 Bε pxqv i pxq dx " ż BεĂT 0 1 σpB ε q v i pxq dx,
and v i is affine (and so harmonic) on T 0 , therefore

`F ε h ˘i " " v i px 0 q if i is a node of the triangle T 0 , 0 otherwise.
We note that F ε h " D h , where D h is the discretized right-hand side vector associated to the Dirac mass. That is why, with A h the Laplacian matrix,

u ε h ´uh " ÿ i node " A ´1 h `F ε h ´Dh ˘‰i v i " 0.
Remark 10. F ε h " D h holds as long as B ε Ă T 0 . Otherwise, we still have u

| Ω 0 " u ε | Ω 0 (Theorem 6), but F ε h ‰ D h , and so u h | Ω ‰ u ε h | Ω .

Proof of Theorem 5

Theorem 5 can now be proved.

Proof. First, by triangular inequality, we can write, for s P t0, 1u:

}u ´uh } s,Ω 0 ď }u ´uε } s,Ω 0 `}u ε ´uε h } s,Ω 0 `}u ε h ´uh } s,Ω 0 .
Besides, thanks to Theorem 6, we have

}u ´uε } s,Ω 0 " 0, (2.5.3)
and thanks to Lemma 6, we have

}u h ´uε h } s,Ω 0 " 0. Finally we get }u ´uh } s,Ω 0 ď }u ε ´uε h } s,Ω 0 . (2.5.4)
We apply Nitsche and Schatz Theorem to e " u ε ´uε h . With ℓ " 2, s " 1, and t " 0,

}e} 1,Ω 0 ď C ph}u ε } 2,Ω 1 `}e} 0,Ω 1 q . (2.5.5)
The domain Ω is smooth and f ε P L 2 pΩq, so u ε P H 2 pΩq Ş H 1 0 pΩq, and then, thanks to Inequality (2.1.1), }e} 0,Ω 1 ď }e} 0,Ω ď Ch 2 }u ε } 2,Ω ď Ch 2 }f ε } 0,Ω .

As }f ε } 0,Ω can be calculated,

}f ε } 0,Ω " ˜żΩ ˆ1 πε 2 1 Bε pyq ˙2 dy ¸1{2 " 1 ε ? π , (2.5.6)
for ε " h{10 (in order to satisfy the assumption on B ε ), we get

}e} 0,Ω 1 ď Ch. ( 2

Limits of the method

This proof cannot be adapted to prove the same rate of convergence for a higher order. Indeed, in dimension 2, numerical results make us expect a result in the form of:

}u ´uh } 1,Ω 0 " Oph k q, (2.5.9)
for a numerical solution of the problem pP δ q using the finite element method P k , k ě 2.

The are two natural ways to adapt the proof to show Inequality (2.5.9):

' we can keep f ε as it is defined in (2.5.1), so that Equation (2.5.3) holds. Nevertheless, the finite elements P k for k ě 2 are not harmonic functions, and so Lemma 6 cannot be used.

' we can use instead of f ε a polynomial function Q k , depending on x 0 , T 0 and k, and supported by T 0 (see [START_REF] Scott | Finite element convergence for singular data[END_REF]). But in this case, Q k is not a radial non negative function of mass 1, and Lemma 5 can not be applied.

None of these adaptative methods permits to prove the expected result (2.5.9).

Numerical illustrations

In this section, we illustrate our theorical results by numerical examples.

Concentration of the error around the singularity. First, we present one of the computations which drew our attention to the fact that the convergence could be better far from the singularity. For this example, we define Ω as the unit disk, Ω " tx " px 1 , x 2 q P R 2 : }x} 2 ă 1u, Ω 0 as the portion of Ω Ω 0 " tx " px 1 , x 2 q P R 2 : 0.2 ă }x} 2 ă 1u, and finally x 0 " p0, 0q the origin. In this case, the exact solution u of problem pP δ q is given by

upxq " ´1 4π log ´x2 1 `x2 2 ¯.
When problem pP δ q is solved by the P 1 -finite element method, the numerical solution u h converges to the exact solution u at the order 1 on the entire domain Ω for the L 2 -norm (see [START_REF] Scott | Finite element convergence for singular data[END_REF]). The previous example has shown that the convergence far from the singularity is faster, since the order of convergence in this case is 2 (see [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF]). The difference of convergence rates for L 2 -norm on Ω and Ω 0 let us suppose that the preponderant part of the error is concentrated around the singularity, as can be seen in Figures 2.5 Estimated orders of convergence. Figure 2.9 shows the order of convergence for L 2 pΩ 0 q-norm for the P k -finite element method, where k " 1, 2, 3 and 4. In this case too, the convergence far from the singularity is the same as in the regular case: the P k -finite element method seems to converge at the order k `1 on Ω 0 for the L 2 -norm. This result has been proved by Köppl and Wohlmuth in [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF].

Figure 2.10 shows the estimated order of convergence for the H 1 pΩ 0 q-norm for the P k -finite element method, where k " 1, 2, 3 and 4, in dimension 2. The convergence far from the singularity (i.e. excluding a neighborhood of the point x 0 ) is the same as in the regular case: the P k -finite element method converges at the order k on Ω 0 for the H 1 -norm, as proved previously with a a | lnphq| multiplier. .10: Estimated order of convergence for H 1 pΩ 0 q-norm for the finite element method P k , k " 1, 2, 3, 4.

Discussion

The three-dimensional case

Dirac mass. The approach presented in this chapter can be extended to the threedimensional case but straighforward adaptations of the proofs lead to a suboptimal result.

In the case of Theorem 3, the solution u belongs to W 1,q 0 pΩq for all q in r1, 3{2r. As a consequence the couple pq ε , q 1 ε q defined in (2.4.5) has to be taken near from p3{2, 3q. For instance,

q ε " 3 2 `ε and q 1 ε " 3 1 ´ε ,
so that, with the same notations, the result of Corollary 1 becomes

}u ´uh } ´k`1,Ω ď Ch k h ´ε´1{2 |u ´uh | 1,qε,Ω .
Moreover, the discrete inf-sup condition in dimension 3 is

inf u h PV k h sup v h PV k h ş Ω ∇u h ¨∇v h }u h } 1,qε,Ω }v h } 1,q 1 ε ,Ω ě Ch ε`1{2 .
Thus when dealing with the estimaye for |u ´uh | 1,qε,Ω , we get

|u ´uh | 1,qε,Ω ď Ch ´ε´1{2 |u| 1,qε,Ω .
Finally, with the asymptotics in 3d }u} 1,qε,Ω "

1 3 ? 4π 1 3 ? 3 ´2q ε 2 ,
we get the estimate

}u ´uh } 1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k´1 3 a | ln h| 2 .
which is clearly suboptimal.

Theorem 5 is also suboptimal in 3d, even if better. Indeed, in 2d or in 3d, the proof readily adapts until the computation of }f ε } 0,Ω , which is in 3d

}f ε } 0,Ω " 1 2 c 3 π 1 ε ?
ε , so that we get

}u ´uh } 1,Ω 0 ď C ? h.
Chapter 2. Local error estimates of the finite element method for the Poisson problem with a Dirac right-hand side Line Dirac along a curve. In 3-dimension, a line Dirac δ Γ along a curve Γ ĂĂ Ω belongs to H ´1´η for all η ą 0, so that the solution u Γ of the Poisson problem with the line Dirac δ Γ belongs to H 1´η . Actually, we have u Γ P W 1,q pΩq for all q P r1, 2r. In this case, with the same notations and assumptions as in Theorem 3, we have the following estimate for u Γ and its Galerkin projection u h Γ ,

}u Γ ´uh Γ } 1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k a | ln h|,
which is quasi-optimal. This result is shown using the same arguments as the ones presented in Section 2.4, but cannot be obtained with the tools given in the proof detailed in [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF].

Dirac mass near the boundary

Theorem 5 excludes some critical cases: Dirac mass should not be closer and closer to the border of the domain Ω. Indeed, for example in the case dpx 0 , BΩq " h 2 , Assumption 4 cannot be satisfied with ε " h{10, but only with ε " h 2 {10. Nevertheless, this small value of ε implies }u ´uh } 1,Ω 0 ď C, so that our method does not even prove the convergence of the approximate solution in this case. Actually, if the distance dpx 0 , BΩq tends to 0, the norm }u} 1,q,Ω , for a fixed 1 ď q ă 2, tends to `8, so that the problem becomes more and more singular. But this question is a completely different problem and should be treated in a different way.

This discussion ends this chapter, devoted to the study of the Poisson problem with a Dirac mass in right-hand side. In next chapter we focus on the analysis of local errors for the finite element method applied to the Stokes problem with a punctual force in source term, and show in this case quasi-optimal convergences on subdomains which exclude the singularity [START_REF] Lacouture | Local error analysis for the stokes equations with a singular source term[END_REF]. The solution of the Stokes problem with a punctual force in source term is not H 1 ˆL2 and therefore the approximation by a finite element method is suboptimal. In the case of the Poisson problem with a Dirac mass in the right-hand side, an optimal convergence for the Lagrange finite elements has been shown on a subdomain which excludes the singularity in L 2 -norm by Köppl and Wohlmuth, and, independently, we have proved a quasi-optimal convergence in H s -norm, for s ě 1 (see Chapter 2). In this chapter, we show a quasi-optimal local convergence in H 1 ˆL2 -norm for a P k {P k´1 finite element method, k ě 2, and for the P 1 b{P 1 finite elements. The error is still analysed on a subdomain which does not contain the singularity. The proof is based on local Arnold and Liu error estimates, a weak version of Aubin-Nitsche duality lemma applied to the Stokes problem and discrete inf-sup conditions. These theoretical results are generalized to a wide class of finite element methods, before being illustrated by numerical simulations.

CHAPTER 3 LOCAL ERROR ANALYSIS FOR THE STOKES EQUATIONS WITH A PUNCTUAL FORCE IN SOURCE TERM

Introduction

This chapter is about the accuracy of the finite element method to solve the Stokes problem with a punctual force in source term. Let us consider this following problem

$ & % ´△u `∇p " δ x 0 F in Ω, divpuq " 0 in Ω, u " 0 on BΩ, (3.1.1)
where Ω Ă R 2 is a square, and δ x 0 F denotes the punctual force F located at x 0 P Ω such that distpx 0 , Ωq ą 0.

Our interest in Problem (3.1.1) is motivated by the modeling of the movement of thin structures in a viscous fluid, such as flagella connected to bacteria or cilia involved in the muco-ciliary transport in the lung [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF]. Indeed, for instance in the lung, the cilium is very thin and a direct simulation with a graded mesh would be too expensive. In the asymptotics of a zero diameter cilium and an infinite velocity, the cilium is thus replaced by a line Dirac of forces in source term. In order to ease the computations, the line Dirac of forces is approached by a sum of punctual forces distributed along the cilium [START_REF] Lacouture | A numerical method to solve the Stokes problem with a punctual force in source term[END_REF]. Finally, by linearity of the Stokes problem, the analysis of the subsequent problem reduces to Problem (3.1.1).

In dimension 2, Problem (3.1.1) has no H 1 pΩq 2 ˆL2 pΩq-solution. Although the numerical solution can be defined in this case, the H 1 pΩq-error (respectively L 2 pΩq-error) for the velocity (respectively the pressure) has no sense, and the L 2 -estimates of the velocity cannot be derived like in the regular case without suitable modifications.

Let us notice that the scalar version of this problem, the Poisson problem with a Dirac mass in right-hand side, has already been widely studied. It occurs in many applications from different areas like in optimal control of elliptic problems with state contraints [START_REF] Casas | Control of an elliptic problem with pointwise state constraints[END_REF] or in the mathematical modeling of electromagnetic fields [START_REF] Jackson | Classical electrodynamics[END_REF]. Problems of this type are found in controllability for elliptic parabolic equations [START_REF] Casas | Parabolic control problems in measure spaces with sparse solutions[END_REF][START_REF] Casas | Spike controls for elliptic and parabolic PDEs[END_REF][START_REF] Leykekhman | Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints[END_REF] and in parameter identification problems with pointwise measurements [START_REF] Rannacher | A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements[END_REF]. In this case, Babuska proved force in source term in [START_REF] Babuška | Error-bounds for finite element method[END_REF] for two-dimension smooth domain an L 2 pΩq-convergence of order h 1´ε , ε ą 0, where h is the mesh size, and Scott has shown in [START_REF] Scott | Finite element convergence for singular data[END_REF] an a priori error estimates of order 2 ´d 2 , where the dimension d is 2 or 3. Casas has got the same result in [START_REF] Casas | L 2 estimates for the finite element method for the Dirichlet problem with singular data[END_REF] for general Borel measures on the right-hand side.

To the best of our knowledge, there is no finite element method convergence result for the Stokes problem with a punctual force in source term. Moreover, in applications, the punctual force (or the Dirac measure) at the point x 0 is often a model reduction approach and the finite element solution does not need to approximate precisely the exact solution at the point x 0 . Thus, it is interesting to estimate the error on a fixed subdomain which does not contain the singularity. In the case of the Poisson problem, Köppl and Wohlmuth recently showed in [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF] a quasi-optimal local convergence for low order in L 2norm for Lagrange finite elements and optimal local convergence for higher orders. A quasi-optimal local convergence in H s -norm, s ě 1 and an optimal local convergence in the case of low order have been proved in dimension 2 in [START_REF] Bertoluzza | Local error estimates of the finite element method for an elliptic problem with a dirac source term[END_REF]. In this chapter, we establish still in dimension 2 local error estimates for the Stokes problem with a punctual force in source term, Problem (3.1.1), and show a quasi-optimal convergence in H 1 ˆL2 -norm. The proof is based on the Arnold and Liu Theorem [4] that establishes local error estimates for finite element discretizations of the Stokes equations with regular source term. It is written in the case of the P k {P k´1 elements for k ě 2, and the MINI finite element method P 1 b{P 1 if k " 1. No graded meshes are required for these results and they imply that there is no pollution effects far from the singularity.

The chapter is organized as follows. Our main result is formulated in Section 3.2 followed by the Arnold and Liu Theorem [4], an important tool for the proof presented in Section 3.3. Our theoretical results are generalized in Section 3.4 and summarized in Section 3.5, before being illustrated in Section 3.6 by some numerical simulations.

Main results

In this section, we first set all the notations used in this chapter. Then, we formulate our main result and give an important tool for the proof: the Arnold and Liu Theorem. For the sake of clarity, this result is first set and proved in the particular case of the P k {P k´1 finite elements (k ě 2) and the P 1 b{P 1 elements. It is generalized in Section 3.4.

Notations

For a domain D, we denote by } ¨}s,q,D (respectively | ¨|s,q,D ) the norm (respectively semi-norm) of the Sobolev space W s,q pDq, and by } ¨}s,D (respectively | ¨|s,D ) the norm (respectively semi-norm) of the Sobolev space H s pΩq. Letters in bold refer to a vector of R 2 or a function with values in R 2 .

For the numerical solution, let us introduce a family of quasi-uniform simplical triangulations T h of Ω, where h is the meshsize. For the approximation spaces V k h and W k h , we

x 0 Ω Ω 0 Ω 1 BΩ 0 BΩ 1 mesh Figure 3.1: Domains Ω 0 and Ω 1 .
use the P k {P k´1 finite elements, for k ě 2, defined as

V k h " v h P C p Ωq 2 ˇˇv h| T P P k rT s, @T P T h ( , W k h " p h P C p Ωq ˇˇp h | T P P k´1 rT s, @T P T h ( ,
and if k " 1, we use the MINI finite element method P 1 b{P 1 , where P 1 b denotes the continous piecewise linear and bubble functions. For a finite element T , the bubble function b is defined by bpxq "

# λ T 1 pxqλ T 2 pxqλ T 3 pxq if x P T, 0 else,
where λ T 1 , λ T 2 and λ T 3 are the barycentric coordinates of x in relation to the triangle T . We fix two subdomains of Ω, called Ω 0 and Ω 1 , such that Ω 0 ĂĂ Ω 1 ĂĂ Ω and x 0 R Ω 1 (see Figure 3.1). We consider a mesh which satisfies the following condition: Assumption 5. For some h 0 , we have for all 0 ă h ď h 0 (see Figure 3.1),

Ω m 0 Ş Ω c 1 " H, where Ω m 0 " ď T PT h T Ş Ω 0 ‰H T.

Statement of our main results

Our main result is given by Theorem 7. The rest of the chapter is mostly concerned by the proof, the generalization and the illustration of this theorem. Theorem 7. Consider Ω 0 ĂĂ Ω 1 ĂĂ Ω satisfying Assumption 5, k ě 1, 1 ď q ă 2, let pu, pq P W 1,q 0 pΩq ˆLq 0 pΩq be the solution of Problem (3.1.1) and pu h , p h q its Galerkin force in source term

projection onto V k h ˆW k h satisfying
ş Ω p h " 0 and ż Ω ∇pu ´uh q :: ∇η ´żΩ pp ´ph qdivpηq " 0 for all η P V k h , ż Ω divpu ´uh qξ " 0 for all ξ P W k h .

(3.2.1)

Under the assumption that pu, pq P H k`1 pΩ 1 q 2 ˆHk pΩ 1 q, there exists h 1 such that if 0 ă h ď h 1 , we have,

}u ´uh } 1,Ω 0 `}p ´ph } 0,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k a | ln h|.

Arnold and Liu Theorem

Before stating Arnold and Liu Theorem, let us enumerate the assumptions that the finite element spaces V k h and W k h have to satisfy so that the theorem is true. Assumption 6. Given two fixed concentric disks B 0 and B with B 0 ĂĂ B ĂĂ Ω, there exists an h 0 such that for all 0 ă h ď h 0 , we have for some integers k 1 and k 2 :

B1 For any 1 ď ℓ, for each v P H ℓ pBq 2 , there exists η P V k h such that }v ´η} 1,B ď Ch r r 1 ´1}v} ℓ,B , r r 1 " minpk 1 `1, ℓq.

For any 0 ď s, for each π P H s pBq, there exists ξ P W k h such that }π ´ξ} 0,B ď Ch r r 2 }π} s,B , r r 2 " minpk 2 `1, sq.

Moreover, if v P H 1 0 pB 0 q 2 (respectively π vanishes on BzB 0 ) then η (respectively ξ) can be chosen to satisfy η P H 1 0 pBq 2 (respectively ξ vanishes on ΩzB). B2 Let ϕ P C 8 0 pB 0 q, v h P V k h and π h P W k h , then there exist η P V k h Ş H 1 0 pBq and ξ P W k h with supp ξ Ă B such that

}ϕv h ´η} 1,B ď Cpϕ, B, B 0 qh}v h } 1,B , }ϕπ h ´ξ} 0,B ď Cpϕ, B, B 0 qh}π h } 0,B .
B3 For each 0 ă h ď h 0 there exists a domain B h with B 0 ĂĂ B h ĂĂ B such that for any 0 ď ℓ, for all v h P V k h and π h P W k h , we have

}v h } 1,B h ď Ch ´1´ℓ }v h } ´ℓ,B h , }π h } 0,B h ď Ch ´ℓ}π h } ´ℓ,B h .
B4 There exists β ą 0 such that for all 0 ă h ď h 0 , there is a domain B h , with B 0 ĂĂ B h ĂĂ B for which

inf π h PW k h supp π h ĂB h sup v h PV k h supp v h ĂB h ş B h divpv h qπ h |π h | 0,B h |v h | 1,B h ě β ą 0.
We now state the following theorem by Arnold and Liu [4], a key tool in the forthcoming proof of Theorem 7.

Theorem (Arnold and Liu [4]). Consider Ω 0 ĂĂ Ω 1 ĂĂ Ω and V k h and W k h satisfy Assumption 6. Suppose that the continuous solution pv, πq P H 1 pΩq 2 ˆL2 pΩq satisfies pv, πq | Ω 1 P H ℓ pΩ 1 q 2 ˆHℓ´1 pΩ 1 q for some ℓ ą 0. Suppose that the finite element solution pv h , π h q P V k h ˆW k h satisfies ş Ω π ´πh " 0 and ż Ω ∇pv ´vh q :: ∇η ´żΩ pπ ´πh qdivpηq " 0 for all η P V k h , ż Ω divpv ´vh qξ " 0 for all ξ P W k h .

Let t be a nonnegative integer. Then there exist a constant C ą 0 and a real h 1 ą 0 depending only on Ω 1 , Ω 0 , and t, such that if 0 ă h ď h 1 we have

}v ´vh } 1,Ω 0 `}π ´πh } 0,Ω 0 ďCph r 1 ´1}v} ℓ,Ω 1 `hr 2 ´1}π} ℓ´1,Ω 1 `}v ´vh } ´t,Ω 1 `}π ´πh } ´t´1,Ω 1 q,
where r 1 " minpk 1 `1, ℓq, r 2 " minpk 2 `2, ℓq, and k 1 , k 2 as in Assumption B1.

Assumptions B1 and B3 are quite standard and satisfied by a wide class of finite element spaces, including all finite element spaces defined on quasi-uniform meshes [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]. The parameters k 1 and k 2 play respectively the role of the order of approximation of the spaces V k h and W k h . In this section, for k ě 2, we have k 1 " k and k 2 " k ´1, and for k " 1, we have k 1 " k 2 " k " 1. Assumption B2 is less common but also satisfied by a wide variety of approximation spaces, including the P 1 b-finite elements [4]. Actually, for Lagrange finite elements, a stronger property than assumption B2 is shown in [START_REF] Bertoluzza | The discrete commutator property of approximation spaces[END_REF]: let 0 ď s ď ℓ ď k, ϕ P C 8 0 pBq and v h P V k h , then there exists η P V k h such that }ϕv h ´η} s,B ď Cpϕqh ℓ´s`1 }v h } ℓ,B .

(3.2.2) Applied for s " ℓ " 1, Inequality (3.2.2) gives assumption B2. When B h " Ω, Assumption B4 is the standard stability condition or discrete inf-sup condition of the Stokes elements. It usually holds as long as B h is a union of elements.

Remark 11. The assumption pv, πq P H 1 pΩq 2 ˆL2 pΩq is not necessary, but it ensures that the finite element solution pv h , π h q is well-defined. In the Dirac right-hand side case, as V k h Ă C pΩq, the discrete solution pu h , p h q is well-defined and Arnold and Liu Theorem holds.

Proof of Theorem 7

This section is devoted to the proof of Theorem 7. We first show a weak version of Aubin-Nitsche duality lemma (Lemma 7), then we establish two discrete inf-sup conditions (Lemmas 8 and 9), and finally we use these results to prove Theorem 7.

Aubin-Nitsche duality lemma with a singular source term

The proof of Theorem 7 is based on Arnold and Liu Theorem. In order to estimate the quantities }u ´uh } ´t,Ω 1 and }p ´ph } ´t´1,Ω 1 , we first show a weak version of Aubin-Nitsche Lemma in the case of the Stokes problem with a singular source term. Lemma 7. Consider the source term f P W ´1,q pΩq 2 " pW 1,q 1 0 pΩq 2 q 1 , 1 ă q ă 2, and let pw, πq P W 1,q 0 pΩq ˆLq pΩq be the unique solution of

$ & % ´△w `∇π " f in Ω, divpwq " 0 in Ω, w " 0 on BΩ.
Let pw h , π h q be the Galerkin projection of pw, πq in V k h ˆW k h . For any integer 0 ď t ď k´1,

}w´w h } ´t,Ω `}π ´πh } ´t´1,Ω ď Ch 2p1{q 1 ´1{2q h t`1 p|w ´wh | 1,q,Ω `|π ´πh | 0,q,Ω q . (3.3.1)
Proof. We aim at estimating, for t ě 0, the H ´tpΩq-norm and the H ´t´1 pΩq-norm respectively of the errors w ´wh and π ´πh :

}w ´wh } ´t,Ω " sup ϕPC 8 0 pΩq 2 1 }ϕ} t,Ω ˇˇˇż Ω pw ´wh q ¨ϕˇˇˇˇ( 3.3.2)
}π ´πh } ´t´1,Ω " sup

ψPC 8 0 pΩq 1 }ψ} t`1,Ω ˇˇˇż Ω pπ ´πh qψ ˇˇˇ( 3.3.3)
The Galerkin projection pw h , π h q satisfies ż Ω π ´πh " 0 and ż Ω ∇pw ´wh q :: ∇η ´żΩ pπ ´πh qdivpηq " 0 for all η P V k h , ż Ω divpw ´wh qξ " 0 for all ξ P W k h .

(3.3.4)

Consider ϕ P C 8 0 pΩq 2 and let pw ϕ , π ϕ q P H t`2 pΩq ˆHt`1 pΩq be the solution of

$ & % ´△w ϕ `∇π ϕ " ϕ in Ω, divpw ϕ q " 0 in Ω, w ϕ " 0 on BΩ.
Existence and uniqueness of the solution to this problem are given in [START_REF] Temam | Navier-Stokes equations[END_REF] (see Chapter I, §2), and we have the estimate

}w ϕ } t`2,Ω `}π ϕ } t`1,Ω ď C}ϕ} t,Ω . (3.3.5)
In dimension 2, by the Sobolev injections established for instance in [START_REF] Brézis | Analyse fonctionnelle : théorie et applications[END_REF], we have

H t`2 pΩq Ă W 1,q 1 pΩq, H t`1 pΩq Ă L q 1 pΩq, (3.3.6)
for all q 1 in r2, `8r. Thus ż Ω pw ´wh q ¨ϕ " ´żΩ pw ´wh q ¨△w ϕ `żΩ pw ´wh q ¨∇π ϕ " ż Ω ∇pw ´wh q :: ∇w ϕ ´żΩ divpw ´wh qπ ϕ .

By adding (3.3.4) in the last equation, we get for any η P V k h and any ξ P W k h , ż Ω pw ´wh q ¨ϕ " ż Ω ∇pw ´wh q :: ∇pw ϕ ´ηq ´żΩ divpw ´wh qpπ ϕ ´ξq `żΩ divpηqpπ ´πh q.

By definition of w ϕ , divpw ϕ q " 0 on Ω, so ż Ω pw ´wh q ¨ϕ " ż Ω ∇pw ´wh q :: ∇pw ϕ ´ηq ´żΩ divpw ´wh qpπ ϕ ´ξq `żΩ divpη ´wϕ qpπ ´πh q ď|w ´wh | 1,q,Ω p|w ϕ ´η| 1,q 1 ,Ω `|π ϕ ´ξ| 0,q 1 ,Ω q `|π ´πh | 0,q,Ω |w ϕ ´η| 1,q 1 ,Ω . Now let us deal with the pressure estimate. For any ψ P C 8 0 pΩq, we denote by r

ψ the function r ψ " ψ ´1 |Ω| ż Ω ψ.
By definition, it is easy to see that r ψ satisfies ż Ω r ψ " 0 and @t ě 0, } r ψ} t`1,Ω ď CpΩq}ψ} t`1,Ω .

We can now establish the result for the pressure: consider ψ P C 8 0 pΩq and the solution pw ψ , π ψ q P H t`2 pΩq ˆHt`1 pΩq of $ & % ´△w ψ `∇π ψ " 0 in Ω, divpw ψ q " r ψ in Ω, w ψ " 0 on BΩ, See [START_REF] Temam | Navier-Stokes equations[END_REF] (Chapter I, §2) for the existence and the uniqueness, and the following estimate

}w ψ } t`2,Ω `}π ψ } t`1,Ω ď C} r ψ} t`1,Ω ď C}ψ} t`1,Ω . (3.3.7)
Moreover,

ż Ω π ´πh " 0, so that ż Ω pπ ´πh qψ " ż Ω pπ ´πh q r ψ `1 |Ω| ż Ω ψ ż Ω π ´πh " ż Ω pπ ´πh q r ψ.
By the Sobolev injections recalled in (3.3.6), and the Galerkin projection property (3.3.4), we can write for all η P V k h , ż Ω pπ ´πh qψ "

ż Ω pπ ´πh q r ψ " ż Ω pπ ´πh qdivpw ψ q " ż Ω
pπ ´πh qdivpw ψ ´ηq `żΩ ∇pw ´wh q :: ∇η.

Then, for all v P W 1,q 0 pΩq ż Ω ∇w ψ :: ∇v ´żΩ π ψ divpvq " 0, so, with v " w ´wh , and for any ξ P W k h , ż Ω pπ ´πh qψ " ż Ω pπ ´πh qdivpw ψ ´ηq `żΩ ∇pw ´wh q :: ∇pη ´wψ q `żΩ π ψ divpw ´wh q " ż Ω pπ ´πh qdivpw ψ ´ηq `żΩ ∇pw ´wh q :: ∇pη ´wψ q `żΩ pπ ψ ´ξqdivpw ´wh q ď|π ´πh | 0,q,Ω |w ψ ´η| 1,q 1 ,Ω `|w ´wh | 1,q,Ω `|w ψ ´η| 1,q 1 ,Ω `|π ψ ´ξ| 0,q 1 ,Ω ˘.

Finally, for any pη 1 , ξ

1 q P V k h ˆW k h , ż Ω pw ´wh q ¨ϕ ď|w ´wh | 1,q,Ω p|w ϕ ´η1 | 1,q 1 ,Ω `|π ϕ ´ξ1 | 0,q 1 ,Ω q `|π ´πh | 0,q,Ω |w ϕ ´η1 | 1,q 1 ,Ω , (3.3.8)
and for any pη 2 , ξ

2 q P V k h ˆW k h , ż Ω pπ ´πh qψ ď|π ´πh | 0,q,Ω |w ψ ´η2 | 1,q 1 ,Ω `|w ´wh | 1,q,Ω `|w ψ ´η2 | 1,q 1 ,Ω `|π ψ ´ξ2 | 0,q 1 ,Ω ˘. (3.3.9)
In order to estimate |w ϕ ´η1 | 1,q 1 ,Ω , |w ψ ´η2 | 1,q 1 ,Ω , |π ϕ ´ξ1 | 0,q 1 ,Ω and |π ψ ´ξ2 | 0,q 1 ,Ω , we need the following result: Proposition 11 (Girault, Raviart, Corollary A.2, page 97 [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]). Let T h be a family of quasi-uniform simplicial triangulations of Ω Ă R 2 , where h is the meshsize. For any 0 ď m ď t `1 ď k, for any mesh element T in the family, for any v P W k`1,q 1 pΩq, any real q 1 ě 2, |v ´Πh v| m,q 1 ,T ď Ch 2p1{q 1 ´1{2q h t`2´m |v| t`2,2,T ,

where Π h v is the P k -interpolant of the function v.

Up to now and until the end of this proof, we take

η 1 " Π h w ϕ and η 2 " Π h w ψ P V k h , ξ 1 " r Π h π ϕ and ξ 2 " r Π h π ψ P W k h ,
where Π h v is the P k -interpolant of the function v and r Π h v is the P k´1 -interpolant of the function v. By (3.3.10), with m " 1, 0 ď t ď k ´1, for all T finite element in the family,

|w ϕ ´η1 | 1,q 1 ,T ď Ch 2p1{q 1 ´1{2q h t`1 |w ϕ | t`2,2,T , (3.3.11
)

|w ψ ´η2 | 1,q 1 ,T ď Ch 2p1{q 1 ´1{2q h t`1 |w ψ | t`2,2,T ,
and with m " 0,

|π ϕ ´ξ1 | 0,q 1 ,T ď Ch 2p1{q 1 ´1{2q h t`1 |π ϕ | t`1,2,T , |π ψ ´ξ2 | 0,q 1 ,T ď Ch 2p1{q 1 ´1{2q h t`1 |π ψ | t`1,2,T .
We denote the triangles of the mesh by tT i u i"1,¨¨¨,N , and we set a " pa i q i and b " pb i q i , where a i " |w ϕ ´η1 | 1,q 1 ,T i and b i " |w ϕ | t`2,2,T i .

By (3.3.11), we have, for all i in rr1, Nss,

a i ď Ch 2p1{q 1 ´1{2q h t`1 b i .
We recall the norm equivalence in R N for 0 ă r ă s,

}x} ℓ s ď }x} ℓ r ď N 1{r´1{s }x} ℓ s ,
with here N " Ch ´2. As 2 ă q 1 , we have }b} ℓ q 1 ď }b} ℓ 2 . Then, we can write

|w ϕ ´η1 | 1,q 1 ,Ω " }a} ℓ q 1 ď Ch t`1 h 2p1{q 1 ´1{2q }b} ℓ q 1 , ď Ch t`1 h 2p1{q 1 ´1{2q }b} ℓ 2 , ď Ch t`1 h 2p1{q 1 ´1{2q |w ϕ | t`2,2,Ω .
Similarly, we get

|w ψ ´η2 | 1,q 1 ,Ω ď Ch t`1 h 2p1{q 1 ´1{2q |w ψ | t`2,2,Ω , |π ϕ ´ξ1 | 0,q 1 ,Ω ď Ch t`1 h 2p1{q 1 ´1{2q |π ϕ | t`1,2,Ω , |π ψ ´ξ2 | 0,q 1 ,Ω ď Ch t`1 h 2p1{q 1 ´1{2q |π ψ | t`1,2,Ω , force in source term
and by (3.3.5) and (3.3.7), we get

|w ϕ ´η1 | 1,q 1 ,Ω ď Ch t`1 h 2p1{q 1 ´1{2q }ϕ} t,2,Ω , |w ψ ´η2 | 1,q 1 ,Ω ď Ch t`1 h 2p1{q 1 ´1{2q }ψ} t`1,2,Ω , |π ϕ ´ξ1 | 0,q 1 ,Ω ď Ch t`1 h 2p1{q 1 ´1{2q }ϕ} t,2,Ω , |π ψ ´ξ2 | 0,q 1 ,Ω ď Ch t`1 h 2p1{q 1 ´1{2q }ψ} t`1,2,Ω ,
Finally, the proof is ended by combining (3.3.2), (3.3.3), (3.3.8), (3.3.9), and the last inequalities.

Corollary 2. Let pu h , p h q P V k h ˆW k h be the Galerkin projection of the solution pu, pq of Problem (3.1.1), for any 0 ă ε ă 1,

}u´u h } ´k`1,Ω `}p ´ph } ´k,Ω ď Ch ´εh k p|u ´uh | 1,qε,Ω `|p ´ph | 0,qε,Ω q ,
where q ε P r1, 2r is defined by

q ε " 2 1 `ε ˆand so q 1 ε " 2 1 ´ε˙. (3.3.12)
Proof. We apply Lemma 7 with f " δ x 0 F, w " u, π " p and t " k ´1. We can explicit Inequality (3.3.1):

2 ˆ1 q 1 ε ´1 2 ˙" 2 ˆ1 ´ε 2 ´1 2 ˙" ´ε, (3.3.13)
and so, it follows

}u´u h } ´k`1,Ω `}p ´ph } ´k,Ω ď Ch ´εh k p|u ´uh | 1,qε,Ω `|p ´ph | 0,qε,Ω q . 3.3.2 Discrete inf-sup conditions in L q ε -norm Section 3.3.3 is devoted to estimate of |u ´uh | 1,qε,Ω and |p ´ph | 0,qε,Ω .
In that prospect, we need to establish two discrete inf-sup conditions.

Lemma 8. With q ε and q 1 ε defined in (3.3.12), the approximation space V k h defined by

V k h " " v h P V k h ˇˇˇż Ω divpv h qp h " 0, @p h P W k h * ,
satisfies the following discrete inf-sup condition:

inf u h P V k h sup v h P V k h ş Ω ∇u h :: ∇v h |u h | 1,qε,Ω |v h | 1,q 1 ε ,Ω ě Ch ε .

Proof of Theorem 7 119

Proof. The bilinear form apu, vq " ż Ω ∇u :: ∇v is continuous and coercive on H 1 0 pΩq, so for V k h vector subspace of H 1 0 pΩq, we have the inf-sup condition:

inf u h P V k h sup v h P V k h ş Ω ∇u h :: ∇v h |u h | 1,Ω |v h | 1,Ω ě α ą 0,
where α only depends on Ω. We recall the following inverse inequality:

Proposition 12 (Ciarlet, Theorem 3.2.6, page 140 [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]). Let T h a family of quasi-uniform simplicial triangulations of Ω Ă R d , where h is the meshsize. For

v h P V k h , 1 ď r, s ă `8, 0 ď ℓ ď m, ˜ÿ T PT h |v h | r m,r,T ¸1{r ď Ch ´drmaxt0,1{s´1{rus h ´pm´ℓq ˜ÿ T PT h |v h | s ℓ,s,T ¸1{s
.

We apply this to any

v h P V k h Ă C pΩq, with d " 2, m " l " 1, s " 2 and r " q 1 ε to get: |v h | 1,q 1 ε ,Ω ď Ch ´2p1{2´1{q 1 ε q |v h | 1,2,Ω " Ch ´ε|v h | 1,2,Ω , using Equation (3.3.13)
. Moreover, for any

u h P V k h , |u h | 1,qε,Ω ď C|u h | 1,2,Ω ď C sup v h P V k h ş Ω ∇u h :: ∇v h |v h | 1,2,Ω ď Ch ´ε sup v h P V k h ş Ω ∇u h :: ∇v h |v h | 1,q 1 ε ,Ω . Finally, inf u h P V k h sup v h P V k h ş Ω ∇u h :: ∇v h |u h | 1,qε,Ω |v h | 1,q 1 ε ,Ω ě Ch ε .
The second discrete inf-sup condition we need is given by the following lemma: Lemma 9. With q ε and q 1 ε defined in (3.3.12), the approximations spaces V k h and W k h satisfy the following discrete inf-sup condition:

inf p h PW k h sup v h PV k h ş Ω divpv h qp h |p h | 0,qε,Ω |v h | 1,q 1 ε ,Ω ě Ch ε .
Proof. The proof is similar to the proof of Lemma 8. By Assumption B4,

inf p h PW k h sup v h PV k h ş Ω divpv h qp h |p h | 0,Ω |v h | 1,Ω ě β ą 0.
By Proposition 12, for any

v h P V k h , |v h | 1,q 1 ε ,Ω ď Ch ´ε|v h | 1,2,Ω .
force in source term So, we have, for any p h P W k h and q ε ă 2,

|p h | 0,qε,Ω ď C|p h | 0,Ω ď C sup v h PV k h ş Ω divpv h qp h |v h | 1,2,Ω ď Ch ´ε sup v h PV k h ş Ω divpv h qp h |v h | 1,q 1 ε ,Ω
.

Finally, we get

inf p h PW k h sup v h PV k h ş Ω divpv h qp h |p h | 0,qε,Ω |v h | 1,q 1 ε ,Ω ě Ch ε . 3.3.3 Estimates of |u ´uh | 1,q ε ,Ω and |p ´ph | 0,q ε ,Ω
Following Corollary 2, the quantities |u ´uh | 1,qε,Ω and |p ´ph | 0,qε,Ω have to be estimated to prove Theorem 7. We apply the last two results to bound them in terms of |u| 1,qε,Ω and |p| 0,qε,Ω .

Lemma 10. Let pu h , p h q P V k h ˆW k h be the Galerkin projection of the solution pu, pq of Problem (3.1.1), for any small enough real ε ą 0,

|u ´uh | 1,qε,Ω ď Ch ´ε p|u| 1,qε,Ω `|p| 0,qε,Ω q .
Proof. First, we estimate |u h | 1,qε,Ω in terms of |u| 1,qε,Ω . As we have divpuq " 0 on Ω, by Equation (3.2.1) we have ż Ω divpu h qq h " 0, @q h P W k h , and so, u h P V k h . By Lemma 8, there exists v h P V k h such as |v h | 1,q 1 ε ,Ω " 1, and

|u h | 1,qε,Ω ď Ch ´ε ż Ω ∇u h :: ∇v h . Moreover, Equation (3.2.1) gives ż Ω ∇u h :: ∇v h " ż Ω ∇u :: ∇v h ´żΩ divpv h qpp ´ph q. Now, v h P V k h , so ż Ω divpv h qp h " 0.
Finally, as |v h | 1,q 1 ε ,Ω " 1, we get

|u h | 1,qε,Ω ď Ch ´ε ˆżΩ ∇u :: ∇v h ´żΩ divpv h qp ˙,
ď Ch ´ε p|u| 1,qε,Ω `|p| 0,qε,Ω q .

We conclude with the triangulary inequality,

|u ´uh | 1,qε,Ω ď |u| 1,qε,Ω `|u h | 1,qε,Ω ď Ch ´ε p|u| 1,qε,Ω `|p| 0,qε,Ω q .
We can now estimate |p ´ph | 0,qε,Ω .

Lemma 11. Let pu h , p h q P V k h ˆW k h be the Galerkin projection of the solution pu, pq of Problem (3.1.1), for any small enough real ε ą 0,

|p ´ph | 1,qε,Ω ď Ch ´2ε p|u| 1,qε,Ω `|p| 0,qε,Ω q .
Proof. The proof is similar to the velocity case: by Lemma 9, there exists

v h P V k h such as |v h | 1,q 1 ε ,Ω " 1 and |p h | 0,qε,Ω ď Ch ´ε ż Ω divpv h qp h .
By Equation (3.2.1), we have ż

Ω divpv h qp h " ´żΩ ∇pu ´uh q :: ∇v h `żΩ divpv h qp.
By applying Lemma 10, as |v h | 1,q 1 ε ,Ω " 1, we get

|p h | 0,qε,Ω ď Ch ´ε ˆ´ż Ω ∇pu ´uh q :: ∇v h `żΩ divpv h qp ˙, ď Ch ´ε p|u ´uh | 1,qε,Ω `|p| 0,qε,Ω q ,
ď Ch ´2ε p|u| 1,qε,Ω `|p| 0,qε,Ω q .

Proof of Theorem 7

We can now prove Theorem 7.

Proof. The functions u and p are analytic on Ω 1 , so the quantities }u} k`1,Ω 1 and }p} k,Ω 1 are bounded. Let us note that in this case pu, pq R H 1 0 pΩq 2 ˆL2 0 pΩq, but Remark 11 allows us to apply Arnold and Liu Theorem. For k 1 " k and

k 2 " " 1 if k " 1, k ´1 if k ě 2,
and l " k `1 " r r 1 and t " k ´1, we have

}u ´uh } 1,Ω 0 `}p ´ph } 0,Ω 0 ď Cph k `}u ´uh } ´k`1,Ω 1 `}p ´ph } ´k,Ω 1 q.
By combining Corollary

}u ´uh } 1,Ω 0 `}p ´ph } 0,Ω 0 ď Ch k a | ln h|,
which ends the proof of Theorem 7.

General case

Theorem 7 and its proof have been written in the particular case of the P k {P k´1 finite element method, k ě 2, and the P 1 b{P 1 elements (which corresponds to the case k " 1).

But we can state two more general results.

Some more general assumptions

First, we focus on the assumptions. Let T h be a family of quasi-uniform simplicial triangulations of Ω, let V k 1 h and W k 2 h be two approximation spaces satisfying Assumption 6. We also assume that V k 1 h P C pΩq: this assumption ensures that the finite element solution is well-defined. Moreover, we need two more assumptions, they play the role of Propositions 11 and 12: Assumption 7. Given B Ă Ω, consider q 1 ě 2, there exists an h 0 such that for all 0 ă h ď h 0 , we have for some positive integers k 1 , k 2 , and some reals R 1 , R 2 ě 1: r B1 For any 0 ď m ď R 1 and m ď ℓ, for each v P H ℓ pBq 2 , there exists η P V k 1 h such that, for any mesh element T Ă B,

|v ´η| m,q 1 ,T ď Ch dp1{q 1 ´1{2q h r r 1 ´m|v| ℓ,2,T , r r 1 " minpk 1 `1, ℓq.
For any 0 ď m ď R 2 and m ď ℓ, for each π P H ℓ pBq, there exists ξ P W k 2 h such that, for any mesh element T Ă B,

|π ´ξ| m,q 1 ,T ď Ch dp1{q 1 ´1{2q h r r 2 ´m|π| ℓ,2,T , r r 2 " minpk 2 `1, ℓq. r B3 For 0 ď m ď ℓ ď R 1 , for all v h P V k 1 h
, for any mesh element T P T h , we have

}v h } ℓ,q 1 ,T ď Ch 2p1{q 1 ´1{2q h m´ℓ }v h } m,2,T .
For 0 ď m ď ℓ ď R 2 , for all π h P W k 2 h , for any mesh element T P T h , we have

}π h } ℓ,T ď Ch m´ℓ }π h } m,T .

General case 123

Assumptions r B1 and r B3 are also satisfied by a wide class of finite element spaces, including all finite element spaces defined on quasi-uniform meshes [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]. They are actually common generalisations of Assumptions B1 and B3. The parameters R 1 and R 2 play the roles of the regularities of the approximation spaces V k 1 h and W k 2 h .

Generalization of Theorem 7

We can now state the following result:

Theorem 8. Consider Ω 0 ĂĂ Ω 1 ĂĂ Ω satisfying Assumption 5, 1 ď q ă 2, the solution pu, pq P W 1,q 0 pΩq 2 ˆLq 0 pΩq of Problem (3.1.1) and pu h , p h q its Galerkin projection onto

V k 1 h ˆW k 2 h satisfying ş Ω p h " 0 and ż Ω ∇pu ´uh q :: ∇η ´żΩ pp ´ph qdivpηq " 0 for all η P V k h , ż Ω divpu ´uh qξ " 0 for all ξ P W k h .
Under the assumption that pu, pq P H k 0 `1pΩ 1 q 2 ˆHk 0 pΩ 1 q, there exists h 1 such that if 0 ă h ď h 1 , we have,

}u ´uh } 1,Ω 0 `}p ´ph } 0,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k 0 a | ln h|,
where k 0 " minpk 1 , k 2 `1q.

Proof. We do not develop the complete proof here because it is essentially the same as the proof of Theorem 7 (see Section 3.3). But we explain two differences between both proofs:

' the result of Lemma 7 holds in this case, but for 0 ď t ď minpk 1 ´1, k 2 q.

' the result of Corollary 2 becomes

}u´u h } ´k0 `1,Ω `}p ´ph } ´k0 ,Ω ď Ch ´εh k 0 p|u ´uh | 1,qε,Ω `|p ´ph | 0,qε,Ω q , (3.4.1)
where k 0 " minpk 1 , k 2 `1q.

The end of the proof is the same.

The next two sections are dedicated to the local error analysis for the Stokes equations with a punctual force in source term in H s ˆHs´1 -norm, with s ě 2. force in source term

Local error estimates in H s ˆHs´1 -norm, with s ě 2

Before proceeding, we have to state the following result. It can be seen as a complement of the Arnold and Liu Theorem.

Theorem 9. Consider Ω 0 ĂĂ Ω 1 ĂĂ Ω, V k 1
h and W k 2 h satisfy Assumption 5. Moreover, we assume that V k 1 h and W k 2 h also satisfy Assumption 6, and Assumption 7 for q 1 " 2. Suppose that pv, πq | Ω 1 P H ℓ pΩ 1 q 2 ˆHℓ´1 pΩ 1 q for some ℓ ą 0, and pv h , π h q P V k h ˆW k h satisfies ş Ω π ´πh " 0 and ż Ω ∇pv ´vh q :: ∇η ´żΩ pπ ´πh qdivpηq " 0 for all η P V k h , ż Ω divpv ´vh qξ " 0 for all ξ P W k h .

Let t be a nonnegative integer. Then, there exist a constant C ą 0 and a real h 1 ą 0 depending only on Ω 1 , Ω 0 , and t, such that for 0 ă h ď h 1 , and 2 ď s ď ℓ, with

s ď k 1 ă R 1 and s ď k 2 `1 ă R 2 `1, we have }v ´vh } s,Ω 0 `}π ´πh } s´1,Ω 0 ďC " h r 1 ´s}v} ℓ,Ω 1 `hr 2 ´s}π} ℓ´1,Ω 1 `h1´s p}v ´vh } ´t,Ω 1 `}π ´πh } ´t´1,Ω 1 q ‰ , (3.4.2)
where r 1 " minpk 1 `1, ℓq, r 2 " minpk 2 `2, ℓq.

We first prove a local version of Theorem 9.

Lemma 12. Suppose the conditions of Theorem 9 are satisfied, then (3.4.2) holds with Ω 0 and Ω 1 replaced by G 0 and G, two concentric disks such as G 0 ĂĂ G ĂĂ Ω 1 .

Proof. Let us fix 2 ď s ď ℓ such as s ď k 1 ă R 1 and s ď k 2 `1 ă R 2 `1, and consider

G 0 ĂĂ G h ĂĂ G 1 ĂĂ G ĂĂ Ω 1
where G 0 , G 1 and G are concentric disks, and G h is a union of mesh elements. Then, for any pη, ξq

P V k 1 h ˆW k 2 h , }v´v h } s,G 0 `}π ´πh } s´1,G 0 ď }v ´η} s,G `}η ´vh } s,G h `}π ´ξ} s´1,G `}ξ ´πh } s´1,G h .
Thanks to Assumption r B3, applied for q 1 " 2,

}η ´vh } s,G h ď h 1´s }η ´vh } 1,G h , }ξ ´πh } s´1,G h ď h 1´s }ξ ´πh } 0,G h , so that we have }v ´vh } s,G 0 `}π ´πh } s´1,G 0 ď }v ´η} s,G `h1´s }η ´vh } 1,G h `}π ´ξ} s´1,G `h1´s }ξ ´πh } 0,G h , ď }v ´η} s,G `}π ´ξ} s´1,G `h1´s p}v ´vh } 1,G 1 `}π ´πh } 0,G 1 q `h1´s p}v ´η} 1,G `}π ´ξ} 0,G q. ( 3.4.3)
We now estimate all the terms in Equation (3.4.3):

• thanks to Assumption r B1, applied for q 1 " 2, there exists η P V k 1 h such that }v ´η} s,G ď Ch r 1 ´s}v} ℓ,G and }v ´η} 1,G ď Ch r 1 ´1}v} ℓ,G , (

where r 1 " minpk 1 `1, ℓq.

• still thanks to Assumption r B1, there exists ξ P W k 2 h such that }π ´ξ} s´1,G ď Ch r 2 ´s}π} ℓ´1,G and }π ´ξ} 0,G ď Ch r 2 ´1}π} ℓ´1,G ,

where r 2 " minpk 2 `2, ℓq.

• applying Arnold and Liu Theorem for Ω 0 " G 1 and Ω 1 " G, we get with Ω 0 " G 0 and Ω 1 " G:

}v ´vh } 1,G 1 `}π ´πh } 0,G 1 ďCph r 1 ´1}v} ℓ,G `hr 2 ´1}π} ℓ´1,G `}v ´vh } ´t,G `}π ´πh } ´t´1,G q. ( 3
}v ´vh } s,G 0 `}π ´πh } s´1,G 0 ďC " h r 1 ´s}v} ℓ,G `hr 2 ´s}π} ℓ´1,G `h1´s p}v ´vh } ´t,G `}π ´πh } ´t´1,G q ‰ .
We can now prove Theorem 9.

Proof. The argument is the same as in Theorem 5.1 in [START_REF] Nitsche | Interior estimates for Ritz-Galerkin methods[END_REF]. Consider d " d 0 {2 where d 0 " distp Ω0 , BΩ 1 q. Cover Ω0 with a finite number of disks G 0 px i q, i " 1, 2, ..., N centered at x i P Ω 0 with diam G 0 px i q " d. Let us note that the number N of disks depends only on Ω 0 and Ω 1 . For all i P rr1, Nss, we define Gpx i q corresponding concentric disks with diam Gpx i q " 2d, so that we have:

Ω0 Ă N ď i"1 G 0 px i q Ă N ď i"1 Gpx i q Ă Ω 1 .
We apply Lemma 12 to each couple pG 0 , Gq, so that we get:

}v ´vh } s,Ω 0 `}π ´πh } s´1,Ω 0 ď N ÿ i"1 }v ´vh } s,G 0 px i q `}π ´πh } s´1,G 0 px i q , ďC N ÿ i"1 " h r 1 ´s}v} ℓ,Gpx i q `hr 2 ´s}π} ℓ´1,Gpx i q `h1´s p}v ´vh } ´t,Gpx i q `}π ´πh } ´t´1,Gpx i q q ‰ , ďC N " h r 1 ´s}v} ℓ,Ω 1 `hr 2 ´s}π} ℓ´1,Ω 1 `h1´s p}v ´vh } ´t,Ω 1 `}π ´πh } ´t´1,Ω 1 q ‰ ,
which ends the proof of Theorem 9.

3.4.4

Local error analysis in H s ˆHs´1 -norm for the Stokes equations with a punctual force in source term, s ě 2

In the same way we proved Theorem 8 based on Arnold and Liu Theorem, we apply Theorem 9 to prove the following result.

Theorem 10. Consider Ω 0 ĂĂ Ω 1 ĂĂ Ω, V k 1 h and W k 2 h satisfy Assumption 5. Moreover, we assume that V k 1 h and W k 2 h also satisfy Assumption 6 and Assumption 7. Consider 1 ď q ă 2, let pu, pq P W 1,q 0 pΩq 2 ˆLq 0 pΩq be the solution of Problem (3.1.1) and pu h , p h q its Galerkin projection onto

V k 1 h ˆW k 2 h satisfying ş Ω p h " 0 and ż Ω ∇pu ´uh q :: ∇η ´żΩ pp ´ph qdivpηq " 0 for all η P V k h , ż Ω divpu ´uh qξ " 0 for all ξ P W k h .
Under the assumption that pu, pq P H k 0 `1pΩ 1 q 2 ˆHk 0 pΩ 1 q, there exists h 1 such that if

0 ă h ď h 1 , we have for 2 ď s ď k 0 ă minpR 1 , R 2 `1q, }u ´uh } s,Ω 0 `}p ´ph } s´1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k 0 `1´s a | ln h|,
where k 0 " minpk 1 , k 2 `1q.

Proof. The proof is similar to the one of Theorem 7. The functions u and p are analytic on Ω 1 , so the quantities }u} k 0 `1,Ω 1 and }p} k 0 ,Ω 1 are bounded. Let us apply Theorem 9 taking ℓ " k 0 `1 and t " k 0 ´1. For 2 ď s ď minpk 1 , k 2 `1q ă minpR 1 , R 2 `1q, we have:

}u ´uh } s,Ω 0 `}p ´ph } s´1,Ω 0 ď C " h k 0 `1´s `h1´s p}u ´uh } ´k0 `1,Ω 1 `}p ´ph } ´k0 ,Ω 1 q ‰ .
Assumptions of Theorem 10 imply that assumptions of Theorem 8 are satisfied. We conclude that Estimate (3.4.1) holds, so that:

}u ´uh } s,Ω 0 `}p ´ph } s´1,Ω 0 ď Ch k 0 `1´s h ´ε" |u ´uh | 1,qε,Ω `|p ´ph | 0,qε,Ω ‰ ,
where q ε is defined in (3.3.12). By combining Lemmas 10 and 11, and Inequalities (1.4.5) and (1.4.6) (see Chapter 1, Section 1.4.1.2), we get

}u ´uh } s,Ω 0 `}p ´ph } s´1,Ω 0 ď Ch k 0 `1´s h ´3ε ? 2 ´qε .
By (3.3.14), with ε " | ln h| ´1, we have

}u ´uh } s,Ω 0 `}p ´ph } s´1,Ω 0 ď Ch k 0 `1´s a | ln h|,
which ends the proof of Theorem 10.

3.5. Summary of the main theoretical results of this chapter 127

Summary of the main theoretical results of this chapter

The aim of this section is to collect the results of this chapter to have a clear view of them. We have proved the following results:

Theorem 11. Consider Ω 0 ĂĂ Ω 1 ĂĂ Ω, V k 1 h Ă C pΩq and W k 2
h satisfy Assumption 5. Moreover, we assume that V k 1 h and W k 2 h also satisfy Assumption 6, and Assumption 7. Consider 1 ď q ă 2, let pu, pq P W 1,q 0 pΩq 2 ˆLq 0 pΩq be the solution of Problem (3.1.1) and pu h , p h q its Galerkin projection onto

V k 1 h ˆW k 2 h satisfying ş Ω p h " 0 and ż Ω ∇pu ´uh q :: ∇η ´żΩ pp ´ph qdivpηq " 0 for all η P V k h , ż Ω divpu ´uh qξ " 0 for all ξ P W k h .
Under the assumption that pu, pq P H k 0 `1pΩ 1 q 2 ˆHk 0 pΩ 1 q, there exists h 1 such that if

0 ă h ď h 1 , for 1 ď s ď k 0 ă minpR 1 , R 2 `1q, we have }u ´uh } s,Ω 0 `}p ´ph } s´1,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k 0 `1´s a | ln h|.
where k 0 " minpk 1 , k 2 `1q.

First, the case s " 1 in Theorem 11 has been proved in the particular context of the P k {P k´1 finite elements, for k ě 2, and the P 1 b{P 1 finite elements (which corresponds to the situation k " 1). It is Theorem 7.

Theorem 11 is a mix of Theorem 8 (case s " 1) and Theorem 10 (case s ě 2), which have been proved in Section 3.4. For s " 1, the proof is based on the Arnold and Liu Theorem [4]. For s ě 2, the result is a consequence of Theorem 9, proved in Section 3.4.3.

Next section is devoted to the numerical illustrations of these theoretical results.

Numerical illustrations

In this section, we present some computations which illustrate the theoretical results proved in this chapter.

Concentration of the error around the singularity. First, we define Ω as the unit square, Ω " r0, 1s 2 . and solve the following Stokes problem with F " t r1, 1s and x 0 " p0.5, 0.5q, Estimated orders of convergence. For this second example, the domain Ω is still the unit square, and Ω 0 is defined as the following portion of Ω, Ω 0 " tx P Ω : }x ´x0 } 2 ą 0.4u , where x 0 " p0.5, 0.5q. We fix F " t r1, 1s and solve Problem 3.1.1 for different mesh sizes h with the P 1 b{P 1 , P 2 {P 1 and P 3 {P 2 finite element methods. Figure 3.10 (respectively Figure 3.11) presents the estimated orders of convergence for the H 1 pΩ 0 q-norm of the velocity (respectively the L 2 pΩ 0 q-norm of the pressure) for these three methods. The convergence far from the singularity (i.e. on Ω 0 ) is the same as in the regular case: the P k {P k´1 method (or the P 1 b{P 1 method if k " 1) converges at the order k on Ω 0 in H 1 -norm for the velocity and in L 2 -norm for the pressure, as proved force in source term Elements P 1 b{P 1 Elements P 2 {P 1 Elements P 3 {P 2 Order = 2.17 in this chapter. Let us just note that there is an over-convergence in pressure for the P 1 b{P 1 elements: the estimated order of convergence is approximately 2, greater than the convergence expected by Theorem 7.

$ & % ´△u `∇p " δ x 0 F in Ω, divpuq " 0 in Ω, u " u δ on BΩ,
About the error in L 2 pΩ 0 q-norm for the velocity, Figure 3.12 suggests that the P k {P k´1 finite element method (or P 1 b{P 1 if k " 1) converges at the order k `1 on Ω 0 . This result has only been observed numerically but it is still an open question.

Discussion

The three-dimensional case

Punctual force. The approach presented in this chapter can be also extended to the three-dimensional case, but like is the case of the Poisson problem, straighforward adaptations of the proofs lead to a suboptimal result. First, the solution pu, pq belongs to W 1,q 0 pΩqˆL q 0 pΩq for all q in r1, 3{2r, as a consequence the couple pq ε , q 1 ε q defined in (3.3.12) has to be once again taken near from p3{2, 3q. We fix Moreover, the discrete inf-sup conditions in dimension 3 are inf

q ε " 3 2 `ε and q 1 ε " 3
u h P V k h sup v h P V k h ş Ω ∇u h ¨∇v h }u h } 1,qε,Ω }v h } 1,q 1 ε ,Ω ě Ch ε`1{2 , and inf p h PW k h sup v h PV k h ş Ω divpv h qp h |p h | 0,qε,Ω |v h | 1,q 1 ε ,Ω ě Ch ε`1{2 .
The rest of the proof is exactly the same, and we finally get the estimate

}u ´uh } 1,Ω 0 `}p ´ph } 0,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k´1 3 a | ln h| 2 .
which is clearly suboptimal.

Line Dirac of force. The proof presented in the case of the punctual force, and the adaptations in 3d suggest that the method lead to a local quasi-optimal convergence in the case where the solution pu, pq belongs to P W 1,q 0 pΩq ˆLq 0 pΩq for all q in r1, 2r. For force in source term example, in 3-dimension, a line Dirac of force along a curve Γ ĂĂ Ω belongs to H ´1´η pΩq for all η ą 0, so that the solution pu Γ , p Γ q of the Stokes problem with the line Dirac of forces δ Γ belongs to P W 1,q 0 pΩq ˆLq 0 pΩq, q P r1, 2r. In this case, we can proof the following estimate

}u Γ ´uΓ h } 1,Ω 0 `}p Γ ´pΓ h } 0,Ω 0 ď CpΩ 0 , Ω 1 , Ωqh k a | ln h|,
which is quasi-optimal. This result is shown using the same arguments as the ones presented in Section 3.3.

Punctual force near the boundary

The conclusions about this critical case are strictly the same as for the Poisson problem. Curious reader is encouraged to see Chapter 2, Section 2.7.2 for further information.

Chapters 2 and 3 establish the analysis of local errors for the finite element methods to solve elliptic problem (Poisson and Stokes problems) with a singular source term. Last chapter presents a new method to solve these problems and recover the optimal convergence rate: it is based on the knowledge of the singularity of the solutions [START_REF] Lacouture | A numerical method to solve the Stokes problem with a punctual force in source term[END_REF]. Once set, this method is applied to the study of mucociliary transport. The modelling of thin structures in a viscous fluid leads to consider elliptic problems with a singular right-hand side: a Dirac mass for the Poisson problem or a punctual force for the Stokes problem. In Chapters 2 and 3, we have established local error estimates for the finite element method. But it is well known that the convergence over the whole domain is suboptimal (see for example [START_REF] Scott | Finite element convergence for singular data[END_REF]). In this chapter, we propose a numerical method which preserves optimality for any approximation order even with a singular source term. It is based on the knowledge of a fundamental solution of the associated operator over the whole space. Unlike Chapters 2 and 3, in which the study is limited to the 2-dimensional case, the method is detailed in dimensions 2 and 3. Besides, it is applied in dimension 3 to the direct simulation of a large forest of cilia. Actually, this method is a performing tool to study muco-ciliary transport efficiency and related diseases: direct simulation allows for example to understand better the influence of some parameters on mucus transport, such as the thickness of periciliary layer (and thus of mucus) or the density of cilia.

CHAPTER 4 SIMULATION IN 3D OF A DENSE FOREST OF CILIA IN A VISCOUS FLUID

A numerical method to solve elliptic problems with a singular source term

As it has been already done in previous chapters, we present the method in the case of the Poisson problem with a Dirac mass right-hand side and in the case of the Stokes problem with a punctual force in source term. The calculations are detailed in 2-dimension only, but the results are also given in dimension 3.

The method is based on the explicit knowledge of the singularity of the solution. The main idea relies on the extraction of the singularity from the solution to reduce initial problem to an auxiliary regular problem. This approach fits on the class of subtraction methods, introduced in [START_REF] Wolters | Numerical mathematics of the subtraction method for the modeling of a current dipole in EEG source reconstruction using finite element head models[END_REF] in the context of electroencephalography.

The Poisson problem with a Dirac mass right-hand side

Even if we aim at applying the method in the case of the Stokes problem, let us start with the Poisson problem for two reasons. First, all the important steps to explain the method are contained in the explanations related to the Poisson problem, and since this problem is the scalar version of Stokes problem, the method is a little simplier in this case. Second, the Poisson problems with a Dirac mass right-hand side appear naturally for example in electromagnetism [START_REF] Wolters | Numerical mathematics of the subtraction method for the modeling of a current dipole in EEG source reconstruction using finite element head models[END_REF]. For the sake of clarity, in all the results of this section the location x 0 of the Dirac mass is the origin. The fundamental solution of the no-boundary condition problem is still denoted u δ , Since a regular lift is sufficient to get the exact solution u (see Chapter 1, Section 1.4.2.2), the singularity of u is contained in the Green's function u δ and is located at the Dirac point x 0 (here, the origin). The following method is based on this observation.

´△u δ " δ 0 in D 1 pR d q. ( 4
In order to extract the singularity, let us fix 0 ă a ă b ă dpx 0 , BΩq and define χ by Definition 1: Definition 1. Assume that χ is a bump function satisfying for some k ě 0, By Definition 1, we can specify u δ χ:

' χ P H 2`k pR d q, ' χ | Bpx 0 , aq " 1,
4.1. A numerical method to solve elliptic problems with a singular source term 139

' on Bpx 0 , aq, χ " 1, so u δ χ " u δ , and therefore g " ´△u δ ´δx 0 " 0.

' on Bpx 0 , bq c , χ " 0, so u δ χ " 0, and therefore g " ´△p0q " 0.

Finally, supppgq Ă R b a px 0 q, where R b a px 0 q is the ring centered around x 0 , of internal radius a and external radius b, see Figure 4.1. Moreover, since u δ is analytic on a neighbourhood of R b a px 0 q and χ P H 2`k pR d q, the function g belongs to H k pΩq.

u h " v h `u0 " v h `uδ χ.
Even if the solutions u and u h do not belong to H 1 pΩq (because of the singularity at the point x 0 ), the difference u ´uh does, and we have }u ´uh } s,Ω " }v ´vh } s,Ω , for 0 ď s ď k `1.

Remark 13. This method allows us to switch from the numerical computation of the solution of a singular problem with Dirac source term (with a poor convergence rate) to the numerical computation of the solution of a regular problem with an optimal convergence rate, at any required precision in terms of regularity.

Practical aspects

First of all, a suitable function χ has to be defined: to take advantage of the use of a P ℓfinite element method, ℓ ě 1, the function χ has to belong to C ℓ pR d q Ş H ℓ`1 pR d q, so that g P H ℓ´1 pΩq, and finally get an optimal convergence. We could choose χ P C 8 0 pR d q, but the calculations to make g be explicit would be unnecessarily complicated. For example, for ℓ " 1, we can take the radial function defined by The expression of g can now be explicited. We detail the calculations only in 2-dimension, but the results are also be given in 3-dimension.

Proposition 13. In dimension d " 2, the function g is given by gpxq " 3 πpb ´aq 3 r `p3r 2 ´2pa `bqr `abq ln r `2r 2 ´2pa `bqr `2ab ˘1R b a px 0 q pxq.

Proof. First, we have already shown that supppgq Ă R b a px 0 q, so for }x} ď a or b ď }x}, gpxq " 0.

For a ă }x} ă b, in polar coordinates, since χ and u δ are radial functions, △pu δ χq " χ△pu δ q `2∇u δ ¨∇χ `uδ △χ

" χ△u δ `2B r pu δ qB r pχq `uδ ˆB2 r,r pχq `1 r B r pχq ˙.
Now, u δ is a solution of Problem (4.1.2), so △pu δ q " 0 on R b a px 0 q. That is why △pu δ χq " 2B r pu δ qB r pχq `uδ Finally, by injecting (4.1.9) into (4.1.8), we get gpxq " ´△pu δ χq " 3 πpb ´aq 3 r `p3r 2 ´2pa `bqr `abq ln r `2r 2 ´2pa `bqr `2ab ˘.

Proposition 14. In dimension d " 3, the function g is given by gpxq " ´3 2πpb ´aq 3 r p2r ´pa `bqq1 R b a px 0 q pxq. 4.1. A numerical method to solve elliptic problems with a singular source term 141

Numerical illustrations

In this section, we illustrate our theorical results by a numerical example. We define Ω the unit disk and x 0 " p0, 0q the origin. Table 4.2 presents the L 2 -error for a direct method and the subtraction method respectively, for characteristic mesh sizes h, and the estimated order of convergence (e.o.c.

The Stokes problem with a punctual force in source term

The previous method has been presented in the case of the Poisson problem. In this section, we adapt the method to the Stokes problem. As in Section 4.1.1, the calculations are detailed in dimension 2 but the results in dimension 3 are also given. Actually, we apply this method in dimension 3, in the case of the Stokes problem, to study muco-ciliary transport.

142 Chapter 4. Simulation in 3D of a dense forest of cilia in a viscous fluid

Principle of the subtraction method

The principle of the method is essentially the same as in the case of the Poisson problem. Let us consider the following problem,

$ & % ´µ△u `∇p " δ x 0 F in Ω, div u " 0 in Ω, u " 0 on BΩ, (4.1.10)
where x 0 is fixed in Ω and F is a vector of R d . For the sake of clarity, we assume x 0 to be the origin. We still denote by pu δ , p δ q the fundamental solution of the no-boundary condition problem x ¨F }x} 3 , (4. 1.12) where I d is the identity matrix in dimension d. The fundamental solution pu δ , p δ q does not satisfy the boundary conditions, and so is not the solution of Problem (4.1.10). But this solution can be retrieved by adding a regular lift, therefore the whole information on the singularity of the solution pu, pq is contained in the fundamental solution pu δ , p δ q and is located at x 0 . In order to extract this singularity, let us fix 0 ă a ă b ă dpx 0 , BΩq and define χ by Definition 1. Then, with u 0 " χu δ and p 0 " χp δ , we define g and h as g " ´µ△u 0 `∇p 0 ´δx 0 F, (4.1.13)

" ´µ△u δ `∇p δ " δ 0 F div u " 0 in D 1 pR d q. ( 4
h " div u 0 . (4.1.14)
By the definitions of u δ , p δ and χ, supppgq Ă R b a px 0 q and suppphq Ă R b a px 0 q, where R b a px 0 q is still the ring centered around x 0 , of internal radius a and external radius b, see Figure 4.1. Since u δ and p δ are analytic on Ωztx 0 u, the regularity of functions g and h directly depends on the regularity of function χ, namely g P H k pΩq and h P H k`1 pΩq. Finally, it only remains to correct the terms u 0 and p 0 by solving the regular elliptic problem $ & % ´µ△v `∇q " ´g in Ω, div v " ´h in Ω, v " 0 on BΩ, and the solution of Problem (4.1.10) is given by pu, pq " pu 0 `v, p 0 `qq " pχu δ `v, χp δ `qq, 4.1. A numerical method to solve elliptic problems with a singular source term 143

where u 0 and p 0 are explicitly known functions and pv, qq is the solution of Problem (4.1.15). Noting pv h , q h q the numerical solution of Problem (4.1.15) and defining u h " v h `u0 and p h " q h `p0 , we have, }u ´uh } H s pΩq " }v ´vh } H s pΩq , for 0 ď s ď k `1, }p ´ph } H s pΩq " }q ´qh } H s pΩq , for 0 ď s ď k.

Again, the method allows us to switch from the numerical computation of the solution of a singular problem with a punctual force in source term (with a poor convergence rate) to the numerical computation of the solution of a regular problem with an optimal convergence rate, at any required precision in terms of regularity.

Practical aspects

We still take (4.1.7) for the choice of χ, and we can establish the expressions of the functions g and h.

Proposition 15. In dimension 2, the functions g and h are given on R b a px 0 q by gpxq " 3 2πpb ´aq 3 r « ˆp3r 2 ´2pa `bqr `abq ln r `2r 2 ´2pa `bqr `2ab ˙I2 `pab ´r2 q x t x r 2 ff F, hpxq " 3p1 ´ln rqpr 2 ´pa `bqr `abq 2πµpb ´aq 3 r x ¨F.

where r " rpxq " }x}, and are zero outside.

Proof. We already know that supppgq Ă R b a px 0 q and suppphq Ă R b a px 0 q. Let us begin with the calculation of h. For a ă }x} ă b, h " divpχu δ q " ∇χ ¨uδ `χdivpu δ q " ∇χ ¨uδ , (4. 1.16) as divpu δ q " 0 on R b a px 0 q, since pu δ , p δ q is solution of Problem (4. 1.11). We need to establish the expression of ∇χ. Noting that B i r "

x i r , with x " t rx 1 , x 2 s, we have Let us now calculate g. As we did previously, g " ´µ△pχu δ q `∇pχp δ q " ´µ pχ△u δ `2∇u δ ∇χ `△χu δ q `χ∇p δ `pδ ∇χ " ´µ p2∇u δ ∇χ `△χu δ q `pδ ∇χ, (4. 1.18) because pu δ , p δ q satisfies ´µ△u δ `∇p δ " 0 on R b a px 0 q. We need to calculate △χ: with Equations (4.1.9), △χ " B 2 r,r pχq `1 r B r pχq " 6p3r 2 ´2pa `bqr `abq pb ´aq 3 r .

B i χ " 6x i pb ´aq 3 r
All that remains for us is to calculate ∇u δ . With F " t rf 1 , f 2 s,

∇u δ " 1 4πµr 2 ˆ" x ¨F x 1 f 2 ´x2 f 1 x 2 f 1 ´x1 f 2 x ¨F  ´2 r 2 x t xF t x ˙.
We can now make all the term in (4.1.18) be explicit: As it has been done in the case of the Poisson problem, we illustrate here our theorical results by a numerical example. We define Ω as the unit square and x 0 " p0.5, 0.5q. Table 4.2 presents the L 2 -error for a direct method and the subtraction method respectively, for characteristic mesh sizes h, and the estimated order of convergence (e.o.c.). Figure 4

´2µ∇u δ ∇χ

Adaptations of the subtraction method to more general problems

In this section, we focus on the advantages and drawbacks of implementing and using the method.

Linear elliptic problems

The subtraction method has been presented in the particular cases of the Poisson and the Stokes problems. Actually, it can be detailed in the case of other linear elliptic problems.

function f is also singular, the method allows the user to separate the difficulties and deal with one singularity after the other.

High number of punctual forces in source term. Let us now consider the Stokes problem with several punctual forces in source term:

$ ' ' ' ' & ' ' ' ' % ´µ△u `∇p " N ÿ i"1 δ x i F i in Ω, divpuq " 0 in Ω, u " 0 on BΩ.
For each punctual force δ x i f i , we define g i and h i as in (4.1.13) and (4. 1.14). No matter the number of punctual forces, the method can be applied and by linearity, only one auxiliary regular problem has to be solved:

$ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % ´µ△u `∇p " ´N ÿ i"1 g i in Ω, divpuq " ´N ÿ i"1 h i in Ω, u " 0 on BΩ.
A high number of punctual forces may involve some bad convergence: even if the source term ř g i (respectively ř h i ) is regular, its L 2 -norm (respectively H 1 -norm) can be very large, and the convergence depends directly on this norm, even if the convergence rate remains the same. Figure 4.5 draws the L 2 -norm of the term source in function of the number of cilia, knowing that a cilium is composed of 20 punctual forces. The L 2 -norm of the source term does not linearly depend on the number of cilia, and even if it can be very large, it stops growing after some number of cilia This screening effect is due to the high density of cilia. The dotted line corresponds to the number of cilia we consider in the simulation, namely 6885 cilia (135 in the direction x and 51 in the direction y). 4.1. A numerical method to solve elliptic problems with a singular source term 149

Note that the required time to numerically build the source term ř g i may become preponderant for a high number of cilia compared to the required time to solve the Stokes problem.

Punctual force close to the border

The issue is the same for the Stokes problem as for the Poisson problem. When presenting the method, we introduce the ring R b a px 0 q centred on x 0 , of internal radius a and external radius b. The natural question is: what happens if the Dirac mass is close to the border of Ω? Indeed, even if we can always define the ring, the mesh size h could be bigger than one of these distances: the radius a, the width of the ring b ´a, the distance of the ring R b a px 0 q to the border BΩ. Moreover, even if h small enough, is the convergence of the finite element solution to the exact solution still optimal? Actually, the convergence is related to the regularity and the norm of the right-hand side g. In our case, we have:

}g} 0,Ω ď }χ} 2,R b a px 0 q }u δ } 1,R b a px 0 q
. Moreover, we can show that there exists C independent of a and b such that

}χ} 2,R b a px 0 q ď C pb ´aq 3 .
As a result, if the distance b ´a is small, the norm }χ} 2,R b a px 0 q is very big (and the growth is cubic). We conclude that the optimal rate of convergence can be counterbalanced by the constant, leading to a deterioration of the results in practice.

A solution to counter-balance this fact is to consider a so-called "Stokeslet in a no-slip boundary", that is to say a Stokeslet defined in the half-space, and satisfying u " 0 on the border (see Figure 4.6). This Stokeslet is built in [START_REF] Blake | A note on the image system for a stokeslet in a no-slip boundary[END_REF]. For the sake of clarity, consider the problem

$ ' & ' % ´µ△r u δ `∇r p δ " δ x 0 F in R 2 ˆR˚, divpr u δ q " 0 in R 2 ˆR˚, r u δ " 0 on the plan tz " 0u,
where F P R 3 and x 0 " px 0 , y 0 , z 0 q, with z 0 ą 0. The solution is the Stokeslet in a no-slip boundary, and it is defined by:

r u δ " 1 8πµ "ˆ1 r ´1 R ˙I3 `X`b X r3 ´X´b X Ŕ3 `2z 0 WpR 1 , R 2 , R 3 q  ¨F, and
r p δ " 1 4π " X r3 ´XŔ 3 ´2z 0 SpR 1 , R 2 , R 3 q  ¨F,
where X `" t rx ´x0 , y ´y0 , z ´z0 s, X ´" t rx ´x0 , y ´y0 , z `z0 s,

r " }X `}, R " }X ´} " b R 2 1 `R2 2 `R2 3 ,
150 Chapter 4. Simulation in 3D of a dense forest of cilia in a viscous fluid

R 1 " x ´x0 , R 2 " y ´y0 , R 3 " z `z0 , WpR 1 , R 2 , R 3 q " » - - - - - - - - - B R 1 ˆz0 R 1 R 3 ´R1 R 3 R 3 ˙BR 2 ˆz0 R 1 R 3 ´R1 R 3 R 3 ˙´B R 3 ˆz0 R 1 R 3 ´R1 R 3 R 3 ḂR 1 ˆz0 R 2 R 3 ´R2 R 3 R 3 ˙BR 2 ˆz0 R 2 R 3 ´R2 R 3 R 3 ˙´B R 3 ˆz0 R 2 R 3 ´R2 R 3 R 3 ḂR 1 ˆz0 R 3 R 3 ´1 R ´R2 3 R 3 ˙BR 2 ˆz0 R 3 R 3 ´1 R ´R2 3 R 3 ˙´B R 3 ˆz0 R 3 R 3 ´1 R ´R2 3 R 3 ˙fi ffi ffi ffi ffi ffi ffi ffi ffi fl , and
SpR 1 , R 2 , R 3 q " » - - - - - - - - - B R 1 ˆR3 R 3 ḂR 2 ˆR3 R 3 ḂR 3 ˆR3 R 3 ˙fi ffi ffi ffi ffi ffi ffi ffi ffi fl " » - - - - - B R 1 B R 2 B R 3 fi ffi ffi ffi ffi fl ˆR3 R 3 ˙.
tz " 0u z y

x px 0 , y 0 , z 0 q F R r px 0 , y 0 , ´z0 q px, y, zq Once we have a fundamental solution, we can apply the subtraction method. To help in this process, we have to define a "good" function χ, but Definition 1 does not match in this case. Let us make another choice for χ: Definition 2. Assume that χ is a bump function satisfying for some k ě 0, χ P H 2`k pΩq. Moreover, let us split the domain Ω in two parts and define χ on each of these parts: 4.1. A numerical method to solve elliptic problems with a singular source term 151

' in the half-space tz ě z 0 u, χ is a radial function centred on x 0 such as χ | Bpx 0 , aq " 1 and χ | Bpx 0 , bq c " 0, as defined in Definition 1.

' in the half-space tz ď z 0 u, χ is a cylindrical function of axis D (the straight line tx " x 0 and y " y 0 u, see Figure 4.7) such as χpx, y, zq " 1 if rpx, yq ď a, χpx, y, zq " 0 if b ď rpx, yq, where rpx, yq " a px ´x0 q 2 `py ´y0 q 2 .

From this point, the method can be applied as in Sections 4.1.1.1 and 4.1.2.1. With this adaptation, we can compute punctual forces δ x 0 F close to the border of Ω, even if an issue persists if the punctual force is near a corner. Note that if we consider periodic boundary conditions, punctual forces can be close to the border (or corner): the part of the support of χ which is out the box is deferred at the other side of the box. Calculations of functions g and h have not been done in this case. Actually, in our model, the cilium is connected to the bottom of the box (domain Ω) and therefore, we should use the Stokeslet in a no-slip boundary. But the lower part of the cilium does not beat a lot, the velocity near the base is low, and so the force F distributed near the border is also very small and vanishes at the border. As a consequence, heavy computations related to the treatment of the near boundary punctual forces can be avoided in the case of mucociliary transport.

Non-constant viscosity

The method has been presented in the case of a constant-viscosity fluid. Actually, this assumption is not really necessary, but the difficulty remains the existence of the fundamental solution. In some cases, we can still define the method with suitable adaptations. For instance, the two-viscosity case is very interesting, as the mucociliary escalator relies on a bifluid mechanism. Consider Ω a viscous fluid domain in 3d of viscosity µ defined by µpx, y, zq "

" µ 1 if z ă h 0 , µ 2 if z ą h 0 ,
where h 0 is the height of the bottom layer. Let us denote by Ω 1 this bottom layer and the top layer by Ω 2 (see Figure 4.8). In applications, the domain Ω 1 corresponds to the periciliary layer while the domain Ω 2 refers to the mucus. Let us assume that the punctual force is located at the point x 0 " px 0 , y 0 , z 0 q with z 0 ă h 0 , so that it is located in the bottom layer, as in Figure 4.8 (else, the development is the same exchanging the subscripts i " 1, 2 for µ i and Ω i ). Let us thus define the Stokeslet pu δ , p δ q as in (4.1.12) with µ " µ 1 , and the functions g and h are given in 3d by Proposition 16, still with µ " µ 1 and χ defined in (4.1.7). Consider u " u 0 `v " χu δ `v and p " p 0 `q " χp δ `q, where pv, pq is the solution of Problem (4. We conclude that pu, pq " pu 0 `r v, p 0 `r qq is the solution of

$ & % ´2divpµDpuqq `∇p " δ x 0 F in Ω, divpuq " 0 in Ω, u " 0 on BΩ.
To complete this adaptated method, it remains to explicit ∇u 0 on Ω 2 :

' on Ω 2 Ş Bpx 0 , bq c , χ " 0, and so ∇u 0 " 0.

' on Ω 2 Ş Bpx 0 , aq, χ " 1, and so ∇u 0 " ∇u δ , with

∇u δ pxq " 1 8πµ 1 r 3 ¨» - - x ¨F xf 2 ´yf 1 xf 3 ´zf 1 yf 1 ´xf 2 x ¨F yf 3 ´zf 2 zf 1 ´xf 3 zf 2 ´yf 3 x ¨F fi ffi fl ´1 r 2 x t xF t x ‹ '.
154 Chapter 4. Simulation in 3D of a dense forest of cilia in a viscous fluid ' on Ω 2 Ş R b a px 0 q, ∇u 0 " ∇pχu δ q " χ∇u δ `uδ t ∇χ, with χ given in (4.1.7), ∇u δ explicited above, and

u δ t ∇χ " 3pr 2 ´pa `bqr `abq 4πµ 1 pb ´aq 3 r 2 ˆI3 `xt x r 2 ˙Ft x.
The method adaptated to the two-viscosity case is used in Section 4.2.4 for the simulation of a forest of cilia and the study of the influences of some parameters, like the thickness of the periciliary layer or the density of cilia.

Application to mucociliary transport in the lung

The complexity of the process is high (cilia, two layers, several interfaces, several scales, etc) and only a few data are available. Although a wide variety of works can be found in the literature about mucociliary clearance, the numerical modeling remains at present time a challenge for the scientists. One of the main difficulties is the model of the cilia. For instance, Smith and coworkers [START_REF] Smith | A viscoelastic traction layer model of muco-ciliary transport[END_REF] replaced the forest by an active porous medium in which the cilia are modeled by a volumic resistive force directly dependent on the local velocity of the cilia. They considered in 2-dimension a three-layer fluid: a Maxwell fluid for the mucus and a Newtonian fluid for the periciliary layer and the layer of transition. This model leads to a quasi-uniform mucus transport but no collective movement in the periciliary layer emerges from this model. Another way to model the cilia is the discretecilia model. Dillon and coworkers [START_REF] Dillon | Fluid dynamic models of flagellar and ciliary beating[END_REF] tried to model the internal elastic and forcegenerating structures of the cilia. They considered in 2D only a few cilia but observed that the interface between the mucus and the periciliary layer stays flat, which justifies the two-layers (sometimes three) models with invariant interfaces. Nevertheless, how the internal ciliary engine affects the ciliary beat form remains an open question. That is why most of the works focus on the flow fields produced by cilia with given beat pattern and frequency. Smith and coworkers [START_REF] Smith | Discrete cilia modelling with singularity distributions : application to the embryonic node and the airway surface liquid[END_REF] considered a discrete-cilia model and used regularized Stokeslet to model in 3D the transport by an infinite array of cilia. Lee and coworkers [START_REF] Lee | Muco-ciliary transport : effect of mucus viscosity, cilia beat frequency and cilia density[END_REF] modeled the cilia by discrete Dirac delta functions distributed all along the cilium. They considered a 2D two-layer flow and the governing equations are solved using the immersed boundary method combined with the projection method. The same techniques were used in 3D by Jayathilake and coworkers [START_REF] Jayathilake | Three-dimensional numerical simulations of human pulmonary cilia in the periciliary liquid layer by the immersed boundary method[END_REF] to simulate a 3D twolayer flow in order to better understand some pathological cases. Finally, Chatelin and Poncet [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF] proposed a 3D model with a variable viscosity: the viscosity is the solution of a convection-diffusion equation. The beat of the cilia is imposed and the effects of the cilia on the fluid are treated by penalization. Most of these approaches are discussed in [START_REF] Smith | Modelling mucociliary clearance[END_REF]. [START_REF] Fulford | Muco-ciliary transport in the lung[END_REF].

The simulations should present the time evolution of flow. In our case, the movement of the cilia is periodic (the period T is 1{f ), and the movement of the cilia is imposed: there is no effect of the fluid on the cilia. Due to the instantaneousness of the Stokes equations, the simulation of one period is sufficient. The time step △t is taken equal to T {20. For the sake of clarity, the time t is not given in seconds but in the form t " n△t, where n is an integer between 0 and 19, and represents the associated step in the period T . In all the simulations presented in this chapter, the number of punctual force composing each cilium is 20. Lastly, we do not present convergence results but we ensured that the computations have indeed converged for the meshsize used in the simulations.

Next sections are devoted to the presentation of the various results obtained using the code CAFES and applying the subtraction method to the mucociliary transport. In particular, this numerical tool allows us to study the influence of some parameters of the mucociliary clearance.

Simulations in a constant-viscosity fluid

In the light of the review of the works related to mucociliary clearance, it is clear that modeling the airway surface liquid as a constant-viscosity fluid is not sufficient. However, comparing the constant-viscosity case to the two-viscosity fluid allows to highlight the impact of the bifluid flow for the mucus transport efficiency. Therefore we start with computing cilia beating in a constant-viscosity fluid.

Simulation of a small patch of cilia

In order to validate the model and the method, we start the computations with the simulation of a small patch of cilia. For this computation, we consider a 20 ˆ10 ˆ10 µm 3 box, discretized by a 64 ˆ32 ˆ32 mesh. The patch of cilia is a 15 ˆ15 array of cilia. Figure 4.9 shows the 3-dimensional flow generated by a patch of cilia in a constant-viscosity fluid at times t " △t s (during the effective stroke) and t " 11△t s (during the recovery stroke). It illustrates the efficiency of transport during the effective stroke compared to a low flow in the opposite direction during the recovery stroke. This observation is a consequence of the non-reversibility of the cilia beat pattern. Note also that the average flow over a period is non-zero.

A whole 3-dimensional forest of cilia is computed in a constant-viscosity fluid in the next section. The parameters are the same as in the previous case, except for the domain and the number of cilia: we consider a 30 ˆ5 ˆ10 µm 3 box, discretized by a 128 ˆ17 ˆ33 mesh. The bottom of the box is filled with cilia, an "infinite" array in each direction modeled by a 153 ˆ51 array of cilia and the biperiodic boundary conditions. The length of the box is exactly the length of one metachronal wave, and the boundary conditions in the directions x and y are biperiodic, so that the flow does not depend on time, up to a translation at the velocity of the metachronal wave. As a consequence, drawing one time step only is sufficient. In the following simulation of a whole forest, the box has the same length as the metachronal wave, and therefore this observation holds.

Simulation of a forest of cilia

Simulations with the bifluid model

The simulation of a constant-viscosity fluid leads to first conclusive observations but is not sufficient for the study of mucociliary transport in the lung. From now, we present simulations with the bifluid model: the periciliary layer and the mucus.

Simulation of a forest of cilia

Let us start with the non-pathological case. Figure 4.11 illustrates the flow produced by a whole forest of cilia in a two-viscosity fluid. All the parameters (among the computational box) are the same as in previous case, but we consider here a bifluid model for which the ratio between the viscosity in the mucus and the viscosity in the periciliary layer is r µ " 50, the ratio measured in the experiments [START_REF] Lee | Muco-ciliary transport : effect of mucus viscosity, cilia beat frequency and cilia density[END_REF][START_REF] Smith | A viscoelastic traction layer model of muco-ciliary transport[END_REF] First of all, note that the mucus, corresponding to the top block in Figure 4.11, is transported at near constant velocity. Note also that the components in y and z are zero in the whole fluid. The velocity in the periciliary layer does not appear to depend on x and y, but only on the height z. It seems to grow linearly from 0 (at the bottom of the domain, against the epithelial lining) to the mucus velocity. There are no recirculations as in the constant-viscosity case. The conclusion is that high viscosity in the mucus allows a more efficient transport. Actually, the viscosity effect is the primary mechanism by which mucus transport is produced. Moreover, mucus is transported at a constant velocity, as a block "sliding" over the periciliary layer, which means it is not transported by fits and starts, and there is no turning back, contrary to the case of a patch of cilia [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF]. This fact results from the collective movement of the high number of cilia. Note that there is no difference between the area in which cilia are in their effective stroke and the area in which they are in their recovery stroke, which is different from the constant-viscosity case (compare Figures 4.10 and 4.11): it is a consequence of the collective dynamics of the cilia.

In Figure 4.12, the velocity of the fluid is drawn as a function of the height of the fluid. This figure confirms that velocity is constant in the mucus and that it linearly grows in the periciliary layer. The simulation allows us to reproduce the profiles described in [START_REF] Matsui | Coordinated clearance of periciliary liquid and mucus from airway surfaces[END_REF][START_REF] Smith | Modelling mucociliary clearance[END_REF].

In what follows, the simulation of the standard case serves as a reference case to study the influence of some parameters and their related pathologies.

Influence of the density of cilia

Now that the non-pathological case has been simulated, we can study the influence of some parameters, and for instance, let us look at the consequences of a low density of cilia. By linearity of the Stokes equations, if we divide the number of cilia by 2 (keeping the same length λ for the metachronal wave), we can expect the same qualitative flow, but with an average transport velocity divided by a factor 2. Nevertheless, it is possible that a low density of cilia causes the loss of the collective dynamics. The aim of the following simulations is to evaluate the limit where the collective dynamics disappear. Figure 4.13 shows the flows produced by lower densities of cilia. Figure 4.13a illustrates the flow when the number of cilia is divided by 2 in each direction of the array of cilia. The total number of cilia is therefore divided by 4. As expected, the average velocity in the mucus is 4 times lower. In the case where the number of cilia is divided by 4 in each direction (Figure 4.13b), the velocity in the mucus is divided by 16, and by 64 when the number of cilia in each direction is divided by 8 (Figure 4.13c). Finally, we conclude that the flow produced by the cilia is proportional to the number of cilia. This fact was forseeable by linearity of the Stokes equations. Note that the collective dynamics hold in these cases.

However, when the number of cilia is divided by 8 in each direction, and therefore a total number of cilia divided by 64, the produced flow is not quite constant over a cilium beat. Figure 4.14 draws the average velocity in the mucus layer at the times t " n△t s, for n P r0, 19s (over one beat). The order of magnitude is significantly the same. The difference in the absolute value comes from the ratio between the number of cilia which are in their effective stroke and the number of cilia which are in their recovery stroke: this ratio is not the same all over the beat, but the variations are small. Actually, this observation is a consequence of a computational artefact: the size of the box related to the metachronal wave length, artefact which disappears when the number of cilia increases.

But, what happens when the number of cilia in each direction is even lower? Let us devide it by 16 for a total number of cilia divided by 256. Figure 4.15 shows the flow generated in this case at two different times of the beat: t " 0 s (Figure 4.15a) and t " 4△t s (Figure 4.15b). At time t " 0 s, there are cilia which penetrate into the mucus, and therefore it can be transported. The velocity in this case follows the proportionnality observed previously. But if there is no cilium which penetrates into the mucus (time t " 4△t s), it is not transported and only a bit of periciliary layer is propelled. In this situation, we lose the collective dynamics and fall in a pathological case. Actually, with the simulation of ciliary activity, the penetration into the mucus is not necessary to obtain ample velocities of propulsion [START_REF] Sleigh | The propulsion of mucus by cilia[END_REF][START_REF] Smith | Modelling mucociliary clearance[END_REF].

As a conclusion of this section, let us remark that the too small length of the computational box does not allow us to bring to light a potential threshold where the collective dynamics disappear. Actually, the box should be of the length of several metachronal waves to hope to observe the loss of collective dynamics, but it involves too expansive computational costs.

Influence of the thickness of the periciliary layer

It is clearly reported in the litterature [START_REF] Sleigh | The propulsion of mucus by cilia[END_REF][START_REF] Smith | Modelling mucociliary clearance[END_REF] that the penetration of cilia into the mucus is very important for its transport: mucus is propelled by cilia during their effective stroke (and only during this stroke) and the penetration accentuates the irreversible nature of the beat of the cilia. Two natural pathological cases emerge:

' the first one is the case where the periciliary layer is too thick, so that the cilia do not reach the mucus layer.

' the second one concerns the case where, on the contrary, periciliary layer is so thin that the cilia reach the mucus not only during their effective stroke but also during their recovery stroke, propulsing the mucus towards the opposite direction.

A too thick periciliary layer. Let us start with studying the first case. We compute a dense forest of cilia with the parameters described above, but assume the periciliary layer is a 6.5 µm thickness layer (against 5 µm in the non-pathological case). For this height of periciliary layer, the cilia cannot reach the mucus and never penetrate into it. As it has been observed experimentally [START_REF] Sleigh | The propulsion of mucus by cilia[END_REF] and recovered numerically [START_REF] Smith | Modelling mucociliary clearance[END_REF], a high ciliary activity does not require penetration of the cilia into the mucus to obtain an efficient mucus transport, even if of course the velocity in the mucus layer is lower. Figure 4.16 shows that this observation is reproduced by our simulations. The robustness of the process results in it: if an overproduction of periciliary liquid occured, so that the cilia could not reach the mucus layer, the periciliary layer and the mucus would be transported and a return to the standard situation could be expected.

We can also take advantage of the thick periciliary layer case to study the influence of the viscosity in the mucus on the transport, that is to say the influence of the ratio of viscosities. In order to do that, we compute the flow for three values of this ratio of viscosities: r µ " 20, r µ " 50 and r µ " 70. Note that r µ " 50 is our reference case, illustrated above (see Figure 4.16). Figure 4.17 draws the mean velocity for each of these three values of the ratio of viscosities, as a function of the height. The flow produced in the periciliary layer is the same in the three cases: the cilia do not see the mucus, and the mucus seems to have no effect on the periciliary layer. Nevertheless, the viscosity in the mucus changes its velocity: the more the mucus is viscous (which corresponds to a high ratio of viscosities r µ ) the more the transport is efficient. Note that the loss of velocity between the cases r µ " 50 and r µ " 20 is only about 5% whereas the ratio is divided by 2.5 (same remark between the cases r µ " 50 and r µ " 70: the gain is about 3.5% for a ratio multiplied by 1.4). The dependence of the transport on the viscosity in the mucus is clearly not linear and not very sensitive. A too thin periciliary layer. Let us now consider the opposite phenomena: a thin periciliary layer. In this situation, the height of the interface mucus-periciliary layer is so low that the cilia reach the mucus layer also in their recovery stroke. The mucus is therefore also propelled in the opposite direction during this phase. Figure 4.18 shows 166 Chapter 4. Simulation in 3D of a dense forest of cilia in a viscous fluid the flow produced in this case. Note that there are big recirculations and very different velocities in the periciliary layer: even if the mucus is propelled also in this case, the transport is not optimal because of the important loss of energy in the periciliary layer. To conclude this section, the average velocity in the top of mucus layer is drawn in Figure 4.20 for different thicknesses of periciliary layer. First of all, let us note that the most efficient case is where the thickness of the periciliary layer is 5 µm, that is the same thickness for periciliary layer and mucus. This configuration corresponds to the non-pathological case: the case selected by the nature is the most efficient. Moreover, Figure 4.20 also shows the robustness of the transport efficiency related to the thickness of the periciliary layer. Nevertheless, the velocity magnitude falls significantly if the thickness of the periciliary is not between 3.5 and 6 µm.

Limits of the model

To end this chapter, we discuss the limits of our model. The first one, and maybe the main one, is that the cilia beat is imposed. Even if most of the works related to mucociliary transport do this assumption, it is not realistic. The second limit is the too high order of magnitude we obtain compared to the experimental data. These differences can be explained by the model, which should be improved.

Imposed beat pattern. In our model, as it is done in nearly all the models (see for instance [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF][START_REF] Lee | Muco-ciliary transport : effect of mucus viscosity, cilia beat frequency and cilia density[END_REF][START_REF] Smith | A viscoelastic traction layer model of muco-ciliary transport[END_REF][START_REF] Smith | Modelling mucociliary clearance[END_REF]), the movement of the cilia is imposed, independently on the environment: no matter if the fluid viscosity is high or low, the cilium beat remains the same. When the cilium penetrates the mucus, its velocity is imposed through the movement, and the cilium beat pattern is not sensitive to the change of viscosity. The resistance of mucus is much higher than the resistance of the periciliary layer, but the movement of the cilium is not affected by this difference of resistance, as if the cilium could adapt its internal force to keep a constant velocity. This assumption is not realistic. Actually, experiments show that the cilium is slowed down by the mucus and ends its beat when it goes out the mucus, as if the mucus had stopped it [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF]. In our model, this is not taken into account, and thus, the internal force of the cilium increases artificially to adapt to the viscosity of the mucus. Indeed, the force is linear in µ and therefore, when the cilium crosses the interface mucus-periciliary layer, the force is multiplied by 50 in the upper part of the cilium.

In order to illustrate this remark, we compute the flow produced by a forest of cilia in a bifluid model with the force generated as if the cilia were in a constant-viscosity fluid (the periciliary layer). Actually, we recall the expression of the force at each point of the cilium (see Chapter 1, Equation (1.3.12) for more details) fps, tq " 2πµ lnpL{r 0 q ˜2I 3 ´9 ξps, tq b 9 ξps, tq } 9 ξps, tq} 2 ¸ucil ps, tq, where µ is the local viscosity at the point ξps, tq. Even if this definition of the force is not consistant with the movement of the cilia, the simulation illustrates the important difference between the standard case computed in Section 4.2.4.1 and a "more realistic case", in which the force but not the movement of the cilia would be imposed. By comparing Figures 4.12 and 4.22, which draw the vertical profiles of velocity, we can see a ratio 27 between the two velocities in the upper part of the mucus (where the velocities are constant). Finally, imposing the movement and deducing the force increases artificially the velocity of the fluid, which is one of the reasons of high orders of magnitude (see below). But even if it is not the most realistic model we can develop, it allows us to compute simulations and study the influence of some parameters on the transport and thus understand the mucociliary clearance. Note that imposing the force rather than the movement of the cilia involves a description of the internal mechanism of the cilium, which is not really well-understood. Moreover, this approach requires the model to take into account the feedback of the fluid on the cilium. Finally, imposing the force is an altogether different model. High order of magnitude. In the non-pathological case, Section 4.2.4.1, the order of magnitude of the predicted velocity in the mucus is 3.5 ˆ10 5 µms ´1, which is 1000 times higher than the experimental measurements: Wanner [START_REF] Wanner | Alteration of tracheal mucociliary transport in airway disease : effect of pharmacologic agents[END_REF] measured rates of 300 µm.s ´1. This difference of order of magnitude can be explained by several reasons related to the model, among them:

' the movement of the cilia is imposed; this first point has been explained in the last paragraph, and we have shown that a loss of a factor 27 can be expected by imposing the force rather than the movement.

' the biperiodic boundary conditions.

' the whole forest is computed.

The biperiodic conditions can also explain the high order of magnitude we obtain with our model. Indeed, the mucus is transported along the bronchi until the end of the trachea, and for a portion of bronchus, the mucus input and output have no reason to be equal. We consider biperiodic boundary conditions because they are natural at the scale of the metachronal wave, and because of the difficulty of imposing realistic conditions of input and output.

Lastly, for most of the simulations presented in this chapter, we compute a whole forest of cilia with no area of inactivity. Actually, as it has been observed by Sanderson and Sleigh [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF], only small areas of cilia are beating at each time, a lot of the cilia being at rest (see Chapter 1, Figures 1.8 and 1.9). By linearity of the Stokes equations, we can expect a higher order of magnitude of the flow produced by a whole forest of cilia than the flow produced by only small areas of ciliary activity. But the study of the collective dynamics generated by several small areas of ciliary activity requires a very big box of computation, which increases significantly the computational costs. That is why the simulation of a whole forest in a box of one-metachronal wave length is a good deal between realistic orders of magnitude and reasonable computational costs.

CONCLUSION ET PERSPECTIVES

L'objectif de la thèse est de modéliser et de simuler de manière directe en 3D le mouvement de cils dans un fluide visqueux, c'est-à-dire prise en compte des mécanismes à l'échelle où ils se produisent, à savoir celle du cil. Il s'agit de décrire en détail le mouvement d'un ou plusieurs cils en interaction avec le fluide environnant, et de développer un outil numérique capable de simuler un très grand nombre de cils battant dans un fluide complexe, cet outil permettant de mieux comprendre le processus de transport mucociliaire et les pathologies associées.

Un des points clés du modèle présenté dans le Chapitre 1 est la façon dont nous avons choisi de prendre en compte les effets des cils sur l'écoulement bifluide. Nous travaillons à mouvement imposé et donc la rétro-action du fluide sur les cils n'est pas prise en considération. Le cil étant un corps très fin et battant à une fréquence très élevée, dans l'asymptotique où le rapport entre épaisseur et longueur tend vers 0, mais à résultante de force constante, il est modélisé par une distribution linéique de forces, elle-même approchée pour des raisons purement numériques par une distribution de forces ponctuelles le long du cil. Cette modélisation conduit à l'étude de problèmes elliptiques singuliers, dont l'analyse numérique est présentée dans le Chapitre 2 pour le problème de Poisson avec une masse de Dirac en second membre, et dans le Chapitre 3 pour le problème de Stokes avec une force ponctuelle en terme source. En particulier, on s'intéresse à des estimations d'erreurs locales, sur des sous-domaines qui ne contiennent pas la singularité. Cette analyse numérique nous a poussé à développer une nouvelle méthode numérique pour la résolution de ces deux problèmes. Basée sur celle des éléments finis, elle s'appuie sur la connaissance explicite d'une solution fondamentale, et permet de retrouver une convergence optimale de la solution approchée vers la solution exacte. Enfin, dans le Chapitre 4, cette méthode est appliquée à l'étude du transport mucociliaire. Elle nous permet notamment de simuler en 3D l'écoulement bifluide généré par toute une forêt de cils. Des pathologies liées à la clairance mucociliaire sont illustrées à travers l'étude de l'influence de certains paramètres telles la hauteur de mucus ou la densité de cil.

Nous terminons ce manuscrit en donnant des pistes naturelles de recherche à explorer pour compléter ce travail.

Analyse numérique

L'analyse numérique des problèmes de Poisson et de Stokes singuliers a soulevé plusieurs problèmes. Le principal étant que les preuves présentées en dimension 2 ne s'étendent pas au cas de la dimension 3 : les résultats qu'on obtient ne sont pas optimaux. Une perspective de ce travail est donc la recherche d'autres techniques que celles présentées et qui pourraient être étendues à la dimension 3.

Dans le cas du problème de Poisson, Köppl et Wholmuth [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF] ont obtenu des estimations d'erreurs locales optimales en norme L 2 pour les éléments finis de Lagrange. Même si les estimations que nous avons obtenues en norme H s , s ě 1, ne sont que quasi-optimales (présence d'un facteur log qui, numériquement, ne se voit pas), elles sont valables pour une très grande classe d'éléments finis : les éléments de Lagrange, de Hermite, les ondelettes, etc. Un des points clés de la preuve de Köppl et Wholmuth [START_REF] Köppl | Optimal a priori error estimates for an elliptic problem with Dirac right-hand side[END_REF] est le résultat de Scott [START_REF] Scott | Finite element convergence for singular data[END_REF] : sur l'ensemble du domaine, il y a convergence en norme L 2 à l'ordre 1 en dimension 2 et 1/2 en dimension 3, contre l'ordre 2 dans le cas régulier. Il n'y a pas de résultat équivalent pour le problème de Stokes : pas de résultat de convergence en norme L 2 pour la vitesse sur l'ensemble du domaine. Un axe de recherche consisterait donc à établir des estimations d'erreur sur l'ensemble du domaine dans ce cas-là aussi, et à les utiliser pour démontrer des estimations d'erreurs locales optimales en normes L 2 et H s , pour s ě 1.

Enfin, la question a été soulevée mais laissée en suspens : que se passe-t-il lorsque la masse de Dirac (ou la force ponctuelle) est proche du bord ? Cette question intéressante change complètement la nature du problème et mérite réflexion car, dans les applications, elle intervient lorsque l'on souhaite simuler la dynamique d'un micro-nageur autopropulsé par un flagelle ou des cils. Dans ce cas, remailler le domaine à chaque itération est coûteux et par ailleurs un nouveau maillage ne suffirait pas à obtenir de bonnes estimations.  , où d est la dimension de l'espace. Comme on connaît une solution en domaine infini, on peut adapter la méthode de soustraction à ce problème défini sur un domaine borné, et pallier les problèmes de convergence liés à la résolution numérique d'un problème singulier. Une extension méthodologique des travaux présentés dans ce manuscrit consisterait donc à dérouler les calculs et à appliquer la méthode, par exemple, à la simulation d'un grand nombre de micro-nageurs dans le but de retrouver des signatures rhéologiques observées expérimentalement [START_REF] Decoene | Microscopic modelling of active bacterial suspensions[END_REF].

Méthode de soustraction

D'autres choix de modélisation

Un modèle viscoélastique pour le mucus. Dans la littérature scientifique consacrée au transport mucociliaire, les auteurs s'accordent sur le fait que la couche périciliaire est essentiellement de l'eau, et a donc un comportement newtonien, même si cette affirmation a récemment été mise en cause par Boucher [START_REF] Boucher | Airway surface dehydration in cystic fibrosis : Pathogenesis and therapy[END_REF]. En revanche, pour le mucus, les modèles proposés peuvent être très différents [START_REF] Chatelin | A hybrid grid-particle method for moving bodies in 3D Stokes flow with variable viscosity[END_REF][START_REF] Craster | Surfactant transport on mucus films[END_REF][START_REF] Mitran | Metachronal wave formation in a model of pulmonary cilia[END_REF]. Pour notre modèle, nous avons choisi de considérer un fluide newtonien, pour la couche périciliaire et pour le mucus. Un autre modèle naturel serait de considérer un modèle bifluide, avec un fluide newtonien pour la couche périciliaire et un fluide viscoélastique pour le mucus, dans l'espoir de retrouver des effets non captés par le modèle bifluide newtonien.

Un modèle diphasique pour le fluide. En s'appuyant sur des expériences de Sanderson and Sleigh [START_REF] Sanderson | Ciliary activity of cultured rabbit tracheal epithelium : beat pattern and metachrony[END_REF] et sur des travaux numériques de Dillon et ses co-auteurs [START_REF] Dillon | Fluid dynamic models of flagellar and ciliary beating[END_REF], nous avons supposé dans notre modèle que l'interface entre le mucus et la couche périciliaire restait plane et invariante. Cette hypothèse peut être relaxée en considérant un modèle diphasique pour le fluide [START_REF] Boyer | A theoretical and numerical model for the study of incompressible mixture flows[END_REF], notamment s'il s'agit de mettre en évidence des phénomènes de régulation de la hauteur de la couche périciliaire. Par exemple, Boyer et ses co-auteurs [START_REF] Boyer | Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model[END_REF]

Application à la nage de micro-organismes

Le travail effectué dans cette thèse et l'outil numérique que nous y avons développé, possèdent un champ d'applications plus large que le transport mucociliaire. Parmi ces applications se trouvent l'étude et la simulation de la nage de micro-organismes. On distingue deux types de micro-nageurs :

' le pusher, qui est propulsé par des flagelles, à l'instar de la bactérie Escherichia Coli ou du spermatozoïde.

' le puller, tracté par des cils, comme la micro-algue Chlamydomonas reinhardtii (voir Figure 4.24).

Dans chacun des deux cas, nous pouvons simuler la nage à l'échelle du cil (ou du flagelle) et espérer retrouver des signatures rhéologiques observées expérimentalement. Pour un fluide de viscosité constante, travailler à mouvement imposé peut aussi être un bon premier modèle, mais il semble alors naturel de prendre en compte la rétro-action du fluide. En particulier, pour des bactéries nageant dans un écoulement donné (cisaillement par exemple), les effets du fluide sur les cils peuvent être prépondérants. Ce travail serait l'occasion de confronter le modèle à des données réelles.

Introduction 1 From

 1 the mucociliary transport to the Stokes problem

Figure 2 :

 2 Figure 2: Bifluide.

Figure 3 :

 3 Figure 3: Modélisation du cil : a. Cil volumique de rayon ε et de force ε ´2, de sorte que la masse totale reste constante quand ε tend vers 0. b. Distribution linéique de forces portée par la courbe Γ du cil. c. Une somme de N forces ponctuelles distribuées le long du cil.

aFigure 4 :

 4 Figure 4: Trajectoire d'un cil pendant un battement. a. Description faite par Sanderson et Sleigh [96]. b. Paramétrisation établie par Fulford et Blake [48].

Figure 5 :

 5 Figure 5: Dessin en 3D d'une forêt de cils (avec ℓ 0 " 2µm au lieu de 0.3µm).

0 Figure 6 :

 06 Figure 6: Illustration of the boundary conditions.

Figure 7 :

 7 Figure 7: Support de χ.

Figure 8 :

 8 Figure 8: Simulation 3D d'une mini forêt. a. Phase effective. b. Phase de retour.

Figure 9 :

 9 Figure 9: Simulation directe en 3D d'une forêt de 6885 cils. a. Affichage avec cils. b. Affichage sans cils.

Figure 1 . 2 :

 12 Figure 1.2: Multi-scales schema of human lung, from the trachea and the first generations of bronchi (» 1 cm) to the epithelial cells (» 5 µm). Modified from [30].

Figure 1 . 3 :

 13 Figure 1.3: Crown of "claws" attached to the tips of the cilium, modified from [105].

Figure 1 . 4 :

 14 Figure 1.4: Diagrams showing the structure of a cilium. a. Structures seen in a longitudinal section. b. Transverse section of the ciliary shaft with the typical arrangement of internal fibrils. c. The arrangement of fibril triplets in the basal body, showing the position of the third subfibril C, and the twist of the peripheral triplets. Modified from Warner [116].

Figure 1 . 5 :

 15 Figure 1.5: The 3D-beat cycle of a tracheal cilium seen from the side and from above. On the left the recovery stroke and on the right the effective stroke. Modified from [105].

Figure 1 . 6 :

 16 Figure 1.6: Mucus and cilia in different situations. Modified from [105].

Figure 1 . 7 :

 17 Figure 1.7: Two adjacent 1-µm-thick sections of cultured rabbit tracheal epithelium examined by scanning electron microscopy. Note the difference in orientation of the cilia during the effective (e) and recovery (r) phases of the beat cycle. Modified from [96].

Figure 1 . 8 :

 18 Figure 1.8: Scanning-electron micrographs of rabbit tracheal ciliated epithelia showing different areas of activity. Modified from [96].

Figure 1 . 9 :

 19 Figure

Figure 1 .

 1 Figure 1.10: A schematic illustration of the envelope model. Modified from [105].

3

 3 .3.1) for the constraint tensor σ. Ω Domain occupied by the fluid u Velocity of the fluid ρ Density of the fluid σ Constraint tensor F e External forces µ Viscosity of the fluid Dpuq Deformation tensor of u p Pressure of the fluid I Identity matrix of M 3 pRq Table 1.2: Notations. 1.3.1.2 Stokes versus Navier-Stokes a. b.

Figure 1 .

 1 Figure 1.11: a. Simmer with a high Reynolds number: Jérémy Stravius at the World Swimming Championships in Barcelona, 2013. b.Swimmer with a low Reynolds number: microalga Chlamydomonas Reinhardtii, picture found and modified from http://www.fytoplankton.cz.

Chapter 1 .

 1 From the mucociliary transport to the Stokes problem1.3.1.3 Effects of the cilia on the fluidIt remains to take into account the interaction between the cilia and the mucus. Let us present two approaches:

2 . 2 Figure 1 . 12 :

 22112 Figure 1.12: Domain Ω in the case of a two-viscosity fluid.

1. 3 .

 3 Mathematical modelling of mucociliary transport 55 defining the viscosity µ by the constant piecewise function µpx, y, zq "

0 Figure 1 . 13 :

 0113 Figure 1.13: Illustration of the boundary conditions.

1. 3 .

 3 Mathematical modelling of mucociliary transport 57 and u ε P H 2 pΩq the associated solution.

Figure 1 . 14 :

 114 Figure 1.14: Three schematic source terms to model a cilium: a. A volumic cilium of small radius ε and force in ε ´2 so that the mass stays constant when ε tends to 0. b. A line distribution of forces. c. A sum of N punctual forces distributed along the cilium.

Figure 1 . 15 :

 115 Figure 1.15: Traces of a cilium during one period of its beat. a. Description made by Sanderson and Sleigh [96]. b. Parametrization established by Fulford and Blake [48].

Figure 1 . 16 :Figure 1 . 17 :

 116117 Figure 1.16: Parametrization of a forest of cilia. a. Another section of cultured rabbit tracheal epithelium examined by scanning electron microscopy, modified from [96]. b.Drawing of a section of a forest with the parametrization established by Fulford and Blake[START_REF] Fulford | Muco-ciliary transport in the lung[END_REF].

1. 3 .

 3 Mathematical modelling of mucociliary transport 65 1.3

Figure 1 .

 1 Figure 1.18: a. The inner expansion: L Ñ `8. b. The outer expansion: r 0 Ñ 0.

  so that the discrete problem (and so the finite element solution) is well-defined. The numerical analysis of such problem is led in Chapter 3. The solution of Problem (1.3.14) is the Stokeslet in a confined domain (in contrast to the Stokeslet defined in the whole domain, Propositions 6 and 7).

Figure 1 . 19 :

 119 Figure 1.19: Stokeslet 2d in a confined domain.

B

  r ϕpr, θq pln r ´ln r ´1q dr ¸dθ ´a ż

Figure 1 . 20 :

 120 Figure 1.20: Solution u of Problem (1.4.7) in 2d and its section ty " 0.5u.

Figure 2

 2 if x P ra, x 0 s, x 0 ´a b ´a pb ´xq if x P rx 0 , bs.

Figure 2 . 1 :

 21 Figure 2.1: Exact solution and numerical solution in dimension 1.

Chapter 2 . 1 meshFigure 2 . 2 :

 2122 Figure 2.2: Domains Ω 0 and Ω 1 .

  vpyq dy " r d´1 σpBBpx, rqq ż S d´1 vpx ´rωq dω, Chapter 2. Local error estimates of the finite element method for the Poisson problem with a Dirac right-hand side thus us prove Theorem 6.

Figure 2 . 3 :

 23 Figure 2.3: Illusration of Theorem 6 in 1D.

Chapter 2 . 4 .Figure 2 . 4 :

 2424 Figure 2.4: Assumption on B ε .

Figure 2 . 5 :Figure 2 . 6 :Figure 2 . 7 :Figure 2 . 8 :

 25262728 Figure 2.5: Error for 1{h » 10.

Chapter 2 . 1 Elements P 2 Elements P 3 Order 10 ´1Figure 2 . 9 : 3 .P 1 Elements P 2 Elements P 3 Elements P 4 Order 10 ´1Figure 2

 2123102931234102 Figure2.9: Estimated order of convergence for L 2 pΩ 0 q-norm for the finite element method P k , k " 1, 2, 3.

Contents 3 . 1

 31 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Statement of our main results . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Arnold and Liu Theorem . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Aubin-Nitsche duality lemma with a singular source term . . . . . . 3.3.2 Discrete inf-sup conditions in L qε -norm . . . . . . . . . . . . . . . . . 3.3.3 Estimates of |u ´uh | 1,qε,Ω and |p ´ph | 0,qε,Ω . . . . . . . . . . . . . . 3.3.4 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 4 . 1

 41 Some more general assumptions . . . . . . . . . . . . . . . . . . . . . 3.4.2 Generalization of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Local error estimates in H s ˆHs´1 -norm, with s ě 2 . . . . . . . . . 3.4.4 Local error analysis in H s ˆHs´1 -norm for the Stokes equations with a punctual force in source term, s ě 2 . . . . . . . . . . . . . . 3.5 Summary of the main theoretical results of this chapter . . . . . 3.6 Numerical illustrations . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  .4.6) Finally, combining (3.4.3), (3.4.4), (3.4.5), and (3.4.6), we get the result of Theorem 9

( 3 .Figure 3 . 8 :Figure 3 . 9 :

 33839 Figure 3.8: Error in pressure, 1{h » 20.

Elements P 1 b{P 1 Elements P 2 {P 1 Elements P 3 {P 2 OrderFigure 3 . 10 :

 12132310 Figure 3.10: Estimated order of convergence for the H 1 pΩ 0 q-norm of the velocity.

Order 10 ´1Figure 3 . 11 :

 10311 Figure 3.11: Estimated order of convergence for the L 2 pΩ 0 q-norm of the pressure.

1 ´ε , Elements P 1 b{P 1 Elements P 2 {P 1 Elements P 3 {P 2 Order 10 ´1Figure 3 . 12 :

 1213210312 Figure3.12: Estimated order of convergence for the L 2 pΩ 0 q-norm of the velocity. so that, with the same notations, the result of Corollary 2 becomes}u´u h } ´k`1,Ω `}p ´ph } ´k,Ω ď Ch ´ε´1{2 h k p|u ´uh | 1,qε,Ω `|p ´ph | 0,qε,Ω q .

Contents

4. 1 .

 1 A numerical method to solve elliptic problems with a singular source term 137

138 Chapter 4 .

 4 Simulation in 3D of a dense forest of cilia in a viscous fluid 4.1.1.1 Principle of the subtraction method First, let us recall the Poisson problem with a Dirac mass in right-hand side " ´△u " δ 0 in Ω, u " 0 on BΩ. (4.1.1)

' χ |Figure 4 . 1 :

 χ41 Figure 4.1: Definition of χ.

1

 1 si r P r0, as, 2r 3 ´3pa `bqr 2 `6abr `b2 pb ´3aq pb ´aq 3 si r P ra, bs, 0 si r ą b, (4.1.7) 140 Chapter 4. Simulation in 3D of a dense forest of cilia in a viscous fluid where the function r is the Euclidean distance from the origin rpxq " }x}.

  Equations (4.1.3) and (4.1.7), ' B r pu δ q " ´1 2πr , ' B r pχq " 6 pb ´aq 3 pr 2 ´pa `bqr `abq, ' B 2 r,r pχq " 6 pb ´aq 3 p2r ´pa `bqq.

Figure 4 . 2 :

 42 Figure 4.2: Section ty " 0u of the error }u´u h } for the direct method and the subtraction method with h " 0.125.

Figure 4 .

 4 2 illustrates the concentration of the error around the singularity, with a wide and infinite peak located at the point x 0 for the direct method.

Figure 4 . 3 :

 43 Figure 4.3: Section ty " 0.5u of the error }u ´uh } for the direct method and the subtraction method with h " 0.125.

Figure 4 . 5 :

 45 Figure 4.5: L 2 -norm of the right-hand side in function of the number of cilia.

Figure 4 . 6 :

 46 Figure 4.6: Construction of the Stokeslet in a no-slip boundary.

Figure 4 . 7 :

 47 Figure 4.7: Definition of χ. Section ty " y 0 u.

Ω 2 , µ 2 Ω 1 , µ 1 x 0 Figure 4 . 8 :

 221048 Figure 4.8: Illustration of the subtraction method for a two-viscosity fluid

Figure 4 . 9 :

 49 Figure 4.9: 3D-Simulation of a patch of cilia in a constant-viscosity fluid. a. Effective stroke (t " △t s). b. Recovery stroke (t " 11△t s).

Figure 4 .

 4 Figure 4.10 illustrates the flow produced by a whole forest of cilia.The parameters are the same as in the previous case, except for the domain and the number of cilia: we consider a 30 ˆ5 ˆ10 µm 3 box, discretized by a 128 ˆ17 ˆ33 mesh. The bottom of the box is filled with cilia, an "infinite" array in each direction modeled by a 153 ˆ51 array of cilia and the biperiodic boundary conditions. The length of the box is exactly the length of one metachronal wave, and the boundary conditions in the directions x and y are biperiodic, so that the flow does not depend on time, up to a translation at the velocity of the metachronal wave. As a consequence, drawing one time step only is sufficient. In the following simulation of a whole forest, the box has the same length as the metachronal wave, and therefore this observation holds.

Figure 4 . 10 :

 410 Figure 4.10: 3D-Simulation of a whole forest of cilia in a constant-viscosity fluid. a. Drawing with the cilia. b. The same flow drawn without the cilia.

Figure 4 .

 4 Figure 4.10 illustrates the mucus transport which emerges from the collective movement of the high number of cilia. Let us note the presence of recirculations over the cilia during their recovery stroke, which results in a loss of energy and therefore a sub-efficiency of the mucus transport, compared to the bifluid case (see below).

  .

Figure 4 . 11 :Figure 4 . 12 :

 411412 Figure 4.11: 3D-Simulation of a whole forest of cilia in a constant-viscosity fluid. a. Drawing with the cilia. b. The same flow drawn without the cilia.

Figure 4 . 13 :

 413 Figure 4.13: Influence of the density on the mucus transport. a. Number of cilia divided by 2 in both directions x and y. b. Number of cilia divided by 4 in both directions x and y. c. Number of cilia divided by 8 in both directions x and y.

Figure 4 . 14 :Figure 4 . 15 :

 414415 Figure 4.14: Velocity in the mucus layer over a beat for a density divided by 8 in each direction.

Figure 4 .

 4 Figure 4.16: 3D-Simulation of the thick periciliary layer case. a. Drawing with the cilia. b. The same flow drawn without the cilia.

Figure 4 . 17 :

 417 Figure 4.17: Variation of the viscosity in the mucus in the case of a too thick periciliary layer (PCL).

Figure 4 . 18 :Figure 4 . 19 :

 418419 Figure 4.18: 3D-Simulation of the thin periciliary layer case. a. Drawing with the cilia. b. The same flow drawn without the cilia.

Figure 4 .

 4 [START_REF] Boyer | Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model[END_REF] draws the average velocity as a function of the height and confirms the ample variations of the velocity in the periciliary layer.Thickness of the periciliary layer in µmVelocity in µm.s ´1

Figure 4 . 20 :

 420 Figure 4.20: Average velocity in the mucus for different thicknesses of periciliary layer.

Figure 4 .

 4 [START_REF] Brokaw | Non-sinusoidal bending waves of sperm flagella[END_REF] shows the flow produced when we keep µ " µ PCL all along the cilium, as if the force generated by the cilium were the same in the periciliary layer and the mucus.

Figure 4 . 21 :

 421 Figure 4.21: Flow generated with the modified force in each point of the upper part of the cilia. a. Drawing with the cilia. b. The same flow drawn without the cilia.

Figure 4 . 22 :

 422 Figure 4.22: Average velocity of the fluid as a function of the height, flow produced by the modified force.

  Comme expliqué dans le Chapitre 4, la méthode dite de soustraction, que nous avons développée et présentée pour la résolution numérique des problèmes de Poisson et de Stokes singuliers, peut être généralisée à d'autres problèmes elliptiques. Mais qu'en est-il si on remplace le terme source par un autre lui aussi singulier ? Par exemple, considérons le problème de Stokes avec en second membre un doublet de Stokes D F , défini parD S " lim εÑ0 δ x 0 F ´δr x 0 F ε ,où F est la force du doublet, x 0 son origine, et r x 0 " x 0 ´εe (dans le cas d'un pusher, voir plus loin). Le vecteur unitaire e est défini par e " F{}F}. Un doublet de Stokes est en fait la limite de la somme de deux forces ponctuelles de même intensité, de même direction mais de sens opposés, lorsque leur distance ε tend vers 0 (voir Figure4.23).

Figure 4 . 23 :

 423 Figure 4.23: Doublet de Stokes.

  Chapter 1. From the mucociliary transport to the Stokes problem 1.3.2 Modeling of a cilium . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.1 Hierarchy of thin-structure models for the cilium . . . . . . 1.3.2.2 Parametrization of a cilium and of a forest of cilia . . . . . 1.3.2.3 Distribution of forces along the cilium . . . . . . . . . . . . 1.3.3 Resulting mathematical problem . . . . . . . . . . . . . . . . . . . . 1.4 Elliptic problems with a singular right-hand side . . . . . . . . . 1.4.1 The Stokes problem with a punctual force in source term . . . . . . 1.4.1.1 The Stokeslet . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.2 Regularity of the solution pu, pq . . . . . . . . . . . . . . . 1.4.1.3 Stokeslet in a confined domain . . . . . . . . . . . . . . . .

	1.1. Some biological data	35
	1.1 Some biological data
	1.1.1 The respiratory system

Contents 1.1 Some biological data . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.1.1 The respiratory system . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.1.2 The mucociliary transport . . . . . . . . . . . . . . . . . . . . . . . . 36 1.1.2.1 The mucus . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.1.2.2 The periciliary layer . . . . . . . . . . . . . . . . . . . . . . 39 1.1.2.3 The cilia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.1.2.4 Some pathologies . . . . . . . . . . . . . . . . . . . . . . . . 46 1.2 Mucociliary transport, state of art . . . . . . . . . . . . . . . . . . 47 1.2.1 Envelope models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.2.2 Sublayer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.2.2.1 Traction-layer model . . . . . . . . . . . . . . . . . . . . . . 48 1.2.2.2 Discrete-cilia model . . . . . . . . . . . . . . . . . . . . . . 49 1.2.3 Non constant viscosity models . . . . . . . . . . . . . . . . . . . . . . 50 1.3 Mathematical modelling of mucociliary transport . . . . . . . . . 51 1.3.1 Modeling of the fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.3.1.1 A Newtonian fluid . . . . . . . . . . . . . . . . . . . . . . . 51 1.3.1.2 Stokes versus Navier-Stokes . . . . . . . . . . . . . . . . . . 52 1.3.1.3 Effects of the cilia on the fluid . . . . . . . . . . . . . . . . 54 1.3.1.4 Bifluid model . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.3.1.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 55 1.4.2 The Poisson problem with a Dirac mass right-hand side . . . . . . . 1.4.2.1 The Green's function . . . . . . . . . . . . . . . . . . . . . 1.4.2.2 Regularity of the solution u . . . . . . . . . . . . . . . . . . 1.4.2.3 Illustration of the solution u . . . . . . . . . . . . . . . . .

Table 1 .

 1

	k " 2 k " 3	449 2.076 ´0.003 0.130 ´0.169 0.054 ´0.072 ´1.502 0.260 ´0.123 0.063 ´0.050 ´0.040 ´0.068 0.007 0.026 0.022 0.010 0.196 0.011 ´0.009 ´1.074 ´0.230 ´0.305 ´0.180 ´0.069 0.001 ´0.080 0.658 0.793 ´0.251 0.049 0.009 0.023 ´0.111 0.381 0.331 0.193 0.082 0.029 0.002 0.048
			b n,k
	k " 1 k " 2 k " 3	n " 1 ´0.030 ´0.093 n " 2 0.080 ´0.044 ´0.017 n " 3 0.037 1.285 ´0.036 ´0.244 ´0.093 ´0.137 n " 4 n " 5 0.062 0.016 ´0.065 n " 6 0.052 0.007 0.051 0.095 ´0.298 0.513 0.004 ´0.222 0.035 ´0.128 ´1.034 0.050 0.143 0.043 0.098 ´0.054 0.210 ´0.367 0.009 0.120 ´0.024 0.102

3: Fourier-least squares coefficients for the cilia beat pattern. The upper and lower numbers in each entry correspond to the x and z components respectively, the y component is always zero.

Table 1

 1

	.1. For a given cilium, s n measures the

where x 0 is a point of the domain Ω. 1.4. Elliptic problems with a singular right-hand side 67

  Proposition 8 gives a particular solution of Problem(1.4.8). We search the solution u δ in the form u δ pxq " Up}x}q, with U : R ˚Ñ R. By the expression of the Laplacian in polar coordinates, we have for any x P R 2 ztp0, 0qu where a and b are two real constants. It is clear that U is defined ignoring an additive contant so that we can choose b " 0. It remains to be determined the constant a: consider ϕ P C 8 0 pR 2 q,

	The solutions of this equation satisfy
		U 1 prq " ae ´lnprq "	a r	,
	and so are
		Uprq " a ln r `b,
	x´△u δ , ϕy " " " "	´żR 2 ´ż 2π 0 ´ż 2π 0 ´a ż 2π u δ pxq△ϕpxqdx ż `8 Uprq△ϕpr, θqrdrdθ 0 ż `8 0 ar ln r ˆB2 r,r ϕpr, θq `1 r	B r ϕpr, θq	θ,θ ϕpr, θq ˙drdθ `1 r 2 B 2
	Proposition 8. The Green's function, defined in 2-dimension by
		u δ pxq "	´1 2π	ln }x},
	is a solution of Problem (1.4.8).
	Proof. △u δ pxq " U 2 p}x}q	`1 }x}	U 1 p}x}q.
		U 2 prq	`1 r	Uprq " 0.

With r " }x}, u δ is a solution of Problem (1.4.8) if and only if U satisfies on R ˚the differential equation 0 ˜"r ln rB r ϕpr, θq 

  Ch k h ´ε|u ´uh | 1,qε,Ω . It remains to estimate the quantity |u ´uh | 1,qε,Ω by bounding |u h | 1,qε,Ω in terms of |u| 1,qε,Ω (Equation (2.4.8)).

	2.4.2 Estimate of |u ´uh | 1,q ε ,Ω

.

4.6)

Finally, for finite elements of order k, with t " k ´1, }u ´uh } ´k`1,Ω ď

  where u 0 is explicitly known and v is the solution of Problem (4.1.6). With v h the finite element solution of Problem (4.1.6), the numerical solution u h of Problem (4.1.1) is set as

			Thus, Problem (4.1.1) is
	reduced to the following regular problem
	"	´△v " ´g in Ω, v " 0 on BΩ,	(4.1.6)
	and the solution u of Problem (4.1.1) is given by
		u " v `u0 " v `uδ χ,

Table 4 .

 4 ). Figure4.2 illustrates the section ty " 0u of the error |u ´uh | in both cases. Numerical simulations evidence the fact that solving the auxiliary problem associated to the subtraction procedure of the singularity is more efficient than solving directly the problem with the punctual force source term. 1: L 2 -error for the direct method and the subtraction method

				h	Direct method Subtraction method
			2 ´3 2 ´4 2 ´5 2 ´6 2 ´7 e.o.c.	1.58 ˆ10 ´2 3.72 ˆ10 ´3 3.89 ˆ10 ´3 1.45 ˆ10 ´3 6.51 ˆ10 ´4 1.06		2.11 ˆ10 4.63 ˆ10 1.22 ˆ10 2.99 ˆ10 8.73 ˆ10 1.98	´3 ´4 ´4 ´5 ´6
	0.01
	0.005										Subtraction method Direct method
	0	-1	-0.8	-0.6	-0.4	-0.2	0	0.2	0.4	0.6	0.8	1

  pr 2 ´pa `bqr `abq,

	By combining (4.1.16) and (4.1.17), we get:
	h "	3pr 2 ´pa `bqr `abq 2πµpb ´aq 3 r	p1 ´ln rqx ¨F.
	and so	∇χ "	6pr 2 ´pa `bqr `abq pb ´aq 3 r	x.	(4.1.17)

Table 4 .

 4 .3 illustrates the section ty " 0.5u of the error |u ´uh | in both cases. Again, numerical simulations evidence the fact that solving the auxiliary problem associated to the subtraction procedure of the singularity is more efficient than solving directly the problem with the punctual force source term. 2: L 2 -error for the direct method and the subtraction method

		h	Direct method Subtraction method
		2 ´3 2 ´4 2 ´5 2 ´6 2 ´7 e.o.c.		1.02 ˆ10 ´2 4.87 ˆ10 ´3 2.36 ˆ10 ´3 1.21 ˆ10 ´3 5.89 ˆ10 ´4 1.02	4.12 ˆ10 1.33 ˆ10 2.92 ˆ10 6.86 ˆ10 2.71 ˆ10 1.88	´3 ´3 ´4 ´5 ´5
	0.01 0.02					Subtraction method Direct method
	0	0	0.2	0.4	0.6	0.8	1

Table 4 .

 4 3: Summary of data for cilia in the lung, from

	Length of cilium	L	6 µm
	Cross-sectional radius	r 0	0.1 µm
	Beat frequency	f	15 Hz
	Cilia spacing	ℓ 0	0.3 µm
	Metachronal wavelength	λ	30 µm

  considèrent le modèle d'Oldroyd pour les fuides viscoélastiques et l'interface diffuse entre les deux fluides est modélisée par le modèle de Cahn-Hilliard. Cependant, ce modèle couplé est non linéaire et n'a pas été étudié avec des termes sources singuliers. De manière générale, se pose la question de comment adapter la méthode de soustraction à des opérateurs non linéaires.Travail à force imposée. Même si travailler à mouvement imposé, comme nous l'avons fait dans la thèse, permet de retrouver des comportements et des champs de vitesses observés expérimentalement et numériquement, on a bien vu dans le Chapitre 4 que ce modèle est limité. Son principal défaut est que la partie haute du cil adapte sa vitesse à la résistance du mucus quand il pénètre dedans. Un modèle plus réaliste serait de non pas imposer le mouvement du cil, mais la force que le cil génère en chaque point sur le fluide. De cette force serait induit le mouvement du cil, et on pourrait alors aussi prendre en compte la rétro-action du fluide sur le cil. Le problème est qu'il n'est pas simple de définir cette force, notamment car les mécanismes internes du cil ne sont pas encore bien compris.

' on immerge les cils dans le fluide, et on traite les cils par pénalisation en posant dans le volume du cil u f luide " u cil .Dans les deux modèles décrits ci-dessus, le maillage doit être très précis au niveau des cils pour pouvoir prendre en compte les effets des cils et leur structure. Comme le cil est très fin, un raffinement de maillage est nécessaire au niveau du cil. Si la simulation de quelques cils en 3D est envisageable, il devient compliqué de simuler toute une forêt de cils. De plus, le premier modèle exige la construction d'un nouveau maillage à chaque pas de temps, ce qui devient vite très coûteux, là où le second modelè autorise un maillage cartésien fixe au cours des itérations en temps. Toujours dans l'objectif de limiter au maximum les coûts de calculs, nous avons choisi de considérer une nouvelle approche. En partant du constat que les cils battent très vite et qu'ils sont très fins, nous nous plaçons dans l'asymptotique où la vitesse du cil tend vers l'infini pendant que le diamètre du cil tend vers 0. À la limite, il reste du cil une distribution linéique de forces, portée par l'axe central du cil. Pour des raisons purement numériques, car il est plus facile d'évaluer une fonction en un point que de l'intégrer le long d'une courbe, nous avons approché cette distribution linéique de forces par une somme de forces ponctuelles distribuées le long du cil (voir Figure3). Les liens entre ces modèles seront discutés dans ce manuscrit. Dans cette étude, on s'intéresse à l'écoulement produit par des cils dont le battement et la fréquence sont imposés.Paramétrisation du cil. Pour compléter ce modèle, il reste à définir les x i , points où sont localisées les forces ponctuelles, ainsi que les c i , intensité de ces forces ponctuelles. En d'autres termes, on doit choisir une paramétrisation du cil et définir la force en chaque point du cil. La paramétrisation que nous avons choisie a été établie par Fulford

2.7.1 The three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 103 2.7.2 Dirac mass near the boundary . . . . . . . . . . . . . . . . . . . . . . 104

3.7.1 The three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 130 3.7.2 Punctual force near the boundary . . . . . . . . . . . . . . . . . . . . 132

Figure 4.24: Chlamydomonas reinhardtii, image modifiée à partir de [119].

Remerciements

force in source term Remark 12. Unlike Problem (3.1.1), Problem (3.6.1) has non homogeneous Dirichlet boundary conditions, but in this case, the exact solution is known: u δ . Thus, it is easier to get some information on the error. 3.7, 3.8 and 3.9 show the repartition of the error on the pressure for the same values of h. In both cases, they illustrate the fact that the error concentrates around the singularity. These simulations made us think that the convergence could be optimal on a subdomain which does not contain the singularity: quasi-optimality has been proved in this chapter (Theorem 7). Consider a linear elliptic operator A, for example of the form

where the coefficients a i,j , b k , c are smooth functions, and assume that we know explicitly a smooth fundamental solution r u δ , that is to say r u δ is smooth except on the point x 0 and satisfies Ar u δ " δ x 0 in D 1 pR d q.

(4. 1.19) Then the subtraction method can be applied to solve the following problem " Au " δ x 0 in Ω, u " 0 on BΩ.

In this example, A is a partial operator of order 2, but the order of derivation does not matter. Moreover, the method can be extended to mixed problems, such as the Stokes problem. In fact, the key points are the linearity of the operator and the explicit knowledge of a fundamental solution smooth out of a compact. Notice that in the general case, it is very difficult to solve explicitly Problem (4. 1.19) and that the method cannot be applied without such a solution.

About boundary conditions

Consider the following problem "

where A is a linear elliptic operator for which there exists a smooth fundamental solution r u δ , and B a boundary condition operator. This operator B is assumed to satisfy the condition Bpv `v0 q " Bpvq (4.1.20)

for any function v 0 such that v 0 " 0 on a neighbourhood of the border BΩ (for instance, Bpv `u0 q " Bv, where u 0 is defined in (4.1.4)). Then the method can be applied in this case and the finite element solution u h is given by

where r u 0 " r u δ χ (Definition 1) and v h is the finite element solution of the following problem " Au " ´g in Ω, Bu " 0 on BΩ, with g defined as in (4.1.5). Note that if B is a linear operator which depends only on the trace on BΩ of u and its derivatives, for example Dirichlet or Neumann boundary conditions, Equation (4.1.20) still holds. Besides, the subtraction method is compatible 4.1. A numerical method to solve elliptic problems with a singular source term 147

with periodic boundary conditions, and these conditions are imposed on the auxiliary problem. Lastly, the method is of course compatible with mixed boundary conditions. For example, in 3-dimension, we can consider biperiodic conditions in directions x and y, homogeneous Dirichlet boundary conditions on the bottom, no-output conditions with no-friction sliding on the top of the box (see Figure 4.4). biperiodic conditions pin x and yq u ¨n " 0 pσ ¨nq ¨τ " 0 u " 0 These boundary conditions are chosen in the modeling of the mucociliary transport in the lung.

More general source terms

In order to clarify the explanations, let us deal with the Stokes problem, even if the principle also extends to other linear elliptic problems concerned by Section 4.1.3.1.

A punctual force plus a smooth function in source term. Let f be a smooth function, and consider the following problem

By linearity, the subtraction method can be applied to get a numerical solution of such problems solving the following auxiliary problem

where the functions g and h are defined in (4.1.13) and (4. 1.14). Note that the regularity of this problem is directly linked to the regularity of the function f. Actually, if the

Implementation

The code CAFES (Cartesian Finite Element Solver) has been developed by Benoit Fabrèges, in collaboration with Loïc Gouarin and Bertrand Maury [START_REF] Fabrèges | A smooth extension method for the simulation of fluid/particles flows[END_REF]. It is written in C/C++ and uses the library PETSc. It works with the 4Q1/Q1-finite element to solve a class of problems for which a solver on a Cartesian grid is needed. The library PETSc is used for the parallel structure of matrices and vectors, as well as the parallel solvers for the linear systems.

The simulations presented below are obtained with the code CAFES, used for the resolution of the regular auxiliary Stokes problem. My contribution to the code is the implementation of the subtraction method, described in Section 4.1, from the construction of the right-hand side to the corrective terms added to the numerical solution of the auxiliary regular problem. This remark leads us to consider the viscosity r µ " 1 in the computations: the velocity flow remains the same, and the pressure is multiplied by the real viscosity µ. In the case of the bifluid model, the observation is the same: it is only necessary to keep the ratio between both viscosities r µ " µ mucus {µ P CL constant.

Choice of the different parameters for the computations

Remark 15. We can take the viscosity we want for the computations because we focus on the flow fields produced by cilia with given beat pattern and frequency. If we would impose the force and not the movement of the cilia, this simplification would not be possible. Table 4.3 recalls the data related to the cilia. These parameters are the ones used in all the simulations (unless otherwise stated). It remains to choose the parameters a and b to complete the definition of χ. Let us take a " d{5 and b " 4d{5, with d defined by d " minpz 0 , h z ´z0 q, where z 0 is the third component of the position of the punctual force, and h z the height of the box of calculations. This choice of a and b is consistant with the biperiodic boundary conditions.