J.-C Aval
email: aval@labri.fr

J.-F Marckert
email: marckert@labri.fr

A SYMMETRY PROPERTY OF SOME HARMONIC ALGEBRAIC CURVES

JEAN-CHRISTOPHE AVAL, JEAN-FRANC ¸OIS MARCKERT

Abstract. The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots z i of a complex polynomial P lie on the unit circle U, the points of U different from the z i , and such that Arg(P (z)) = θ, form a regular n-gon, where n is the degree of P .

Let z = {z 1 , . . . , z n } be a multiset of n points in the complex plane C and P the monic polynomial with root set z:

P (z) = n i=1 (z -z i ).
For θ a fixed real number of your choice, consider

C θ (P ) = {z ∈ C : Im(e -iθ P (z)) = 0}.
The set C θ (P ) coincides up to z, to the set {z ∈ C : Arg(P (z)) = θ[π]}. These curves arise in the Gauss approach to the Fundamental Theorem of Algebra (see e.g. Stillwell [START_REF] Stillwell | Mathematics and its history[END_REF], and Martin & al. [START_REF] Martin | Harmonic algebraic curves and noncrossing partitions[END_REF]). In their paper Martin & al. [START_REF] Martin | Harmonic algebraic curves and noncrossing partitions[END_REF] and then Savitt [START_REF] Savitt | Polynomials, meanders, and paths in the lattice of noncrossing partitions[END_REF] initiated the study of the combinatorial topology of the families C θ (P ). The idea are the following ones: the curves C θ (P ) have 2n asymptotes at angles (πk + θ)/n, for k ∈ {0, . . . , 2n -1}, and form in the generic case n non intersecting curves. This induces a matching: k and k ′ are matched if and only if the asymptotes (πk + θ)/n and (πk ′ + θ)/n lie on the same connected component in C θ (P ). The papers [START_REF] Martin | Harmonic algebraic curves and noncrossing partitions[END_REF] and [START_REF] Savitt | Polynomials, meanders, and paths in the lattice of noncrossing partitions[END_REF] aim at studying these matchings, and also the properties of the so-called necklaces, formed by the families of matchings obtained when θ traverses the set [0, π].

Let us now state and prove our result. The set z is clearly included in C θ (P ). It turns out that when z is included in the unit circle U = {z : |z| = 1}, the set C θ (P ) ∩ U presents a quite surprising symmetry -illustrated at Figure 1 -that can be stated as follows.

Proposition 1. If z is a subset of U, then C θ (P ) ∩ U = z ∪ G(z)
where G(z) is the regular n-gon on U, with set of vertices e i(Ω+2kπ/n) , k = 1, . . . , n , for

Ω := 2θ -n j=1 Arg(z j ) n -π.
There exists a purely geometric proof of this Proposition using that the measure of a central angle is twice that of the inscribed angle intercepting the same arc; we provide below a more compact analytic proof.

This work has been supported by the ANR project MARS (BLAN06-2 0193). Proof. We will only consider z / ∈ z. We have the equivalence:

z ∈ C θ (P ) \ z ⇐⇒ z / ∈ z, n i=1 Arg(z -z i ) = θ [π],
where Arg(z) ∈ R/2πZ stands for (any chosen determination of) the argument of z = 0. Now for any ν and ψ real numbers,

e iν -e iψ = e i ν+ψ 2 e i ν-ψ 2 -e i -ν+ψ 2 = 2i sin((ν -ψ)/2)e i ν+ψ 2 . Thus Arg(e iν -e iψ ) = ν + ψ 2 + π 2 + π × sgn sin((ν -ψ)/2) [2π] Hence, z ∈ C θ (P ) \ z is equivalent to: z / ∈ z, n j=1 Arg(z) + Arg(z j ) 2 + π 2 = θ [π],
which leads to the conclusion at once. Note. If z i is a root of multiplicity k of P , and if z i belongs to G(z), then in the neighborhood of z i , C θ (P ) has k tangents, one of them coinciding with the tangent of the circle at z i . Moreover, it is simple to check that if z i is not on G(z), then the tangents of C θ (P ) at z i are not tangent to U.

1

 1

Figure 1 .

 1 Figure 1. An example where n = 7, θ = 0 and the roots z i randomly chosen.


8h00 -8h30 Entrega de Material e CracháCredencial and Materials 8h30 -9h00 AberturaOpening Session Dr. Franco Maria Lajolo (FCF-USP/FoRC), Dra. Bernadette D. G. M. Franco (FCF-USP/Forc) e Dra. Neuza M. A. Hassimotto (FCF-USP/FoRC) Painel 1: Evidências e mecanismoEvidence and Mechanisms CoordenaçãoCoordinator: Profa. Neuza Hassimotto (FCF-USP) 9h00 -9h30 Perspectiva de consumo do suco de laranja no Brasil e no mundo Orange juice consumption-Worlwide Perspective A confirmar (CitrusBR) 9h30 -10h20 Evidências dos efeitos dos polifenóis da laranja na saúde Evidences and research strategies of Citrus polyphenols on health benefits Dr. Francisco A. Tomás Barberán (CEBAS-CSIC, Spain) 10h20 -10h30 Q&A 10h30 -10h50 Coffee Break 10h50 -11h20 Frutas cítricas e modulação da resposta inflamatória Citrus fruits and modulation of the inflammatory response Dr. Marcelo Rogero (Faculdade de Saúde Pública-USP/FoRC) 11h20 -11h50 Potencial farmacológico de Citrus polimetoxiflavonas Pharmacological opportunities with citrus polymethoxyflavones Dra. Thais Borges Cesar (Unesp-Araraquara) 11h50 -12h00 Q&A AlmoçoLunch Poster Session Painel 2: Evidências e MecanismosEvidence and Mechanism CoordenaçãoCoordinator: Dr. Franco Maria Lajolo (FCF-USP/FoRC) 14h00 -14h40 Biodisponibilidade e biomarcadores da ingestão de Flavanonas Bioavailability and Biomarkers of flavanones intake Dr. Paul Kroon (Quadram Institute-UK) 14h40 -15h10 Diversidade do microbioma intestinal: Modulação pelo consumo da laranja Gut Microbiome diversity: Modulation by orange juice consumption Dra. Neuza Mariko Aymoto Hassimotto (Faculdade de Ciências Farmacêuticas-USP/FoRC) 15h10 -15h20 Q&A 15h20 -15h40 Coffee Break Local: Hotel Maksoud Plaza, SP 22 e 23 de de 2018 * * 15h40 -16h20 Orange juice and Cardiovascular disease Orange juice and cardiovascular disease Dr. Dragan Milenkovic (INRA, France) 16h20 -16h50 Influência do consumo de suco de laranja na remodelação cardíaca após o infarto do miocárdio Influence of orange juice intake on cardiac remodeling after myocardial infarct Dr. Sergio Alberto Rupp de Paiva (Unesp-Botucatu/FoRC) 16h50 -17h00 Q&A 23 de Março  March 23 rd Painel 3: Citrus e NutriçãoCitrus and Nutrition CoordenaçãoCoordinator: Prof. Thomas Ong (FCF-USP) 9h00 -9h30 A importância dos Citrus na alimentação Citrus intake -Nutritional significance Dra. Silvia Cozzolino (Faculdade de Ciências Farmacêuticas-USP) 9h30 -10h00 Laranja e Nutrição Infantil Citrus intake and children nutrition Dr. Rubens Feferbaun (Faculdade de Medicina-USP) 10h00 -10h10 Q&A 10h10 -10h30 Coffee Break 10h30 -11h10 Flavonoides de Citrus e fun;ão cognitiva-mecanismos moleculares de sua ação Impact of Citrus flavonoids on cognitive functions and their underlying molecular mechanisms of actions Dr. David Vauzour (University of East Anglia, UK) 11h10 -11h40 Consumo de Citrus durante o início da vida-focus em epigenética Consumption of Citrus during early-life and long term health-focus on epigenetics Dr. Thomas Prates Ong (Faculdade de Ciências Farmacêuticas-USP/FoRC) 11h40 -11h50 Q&A AlmoçoLunch Poster Session Painel 4: Citrus e NutriçãoCitrus and Nutrition CoordenaçãoCoordinator: Dr. Franco Maria Lajolo (FCF-USP/FoRC) variedades de Citrus: composição e bioacessibilidade Carotenoids in different varieties of Citrus: composition and bioaccessibility Dra. Adriana Zerlotti Mercadante (Faculdade de Engenharia de Alimentos-Unicamp/FoRC) Dr. Neuza M.A. Hassimotto/ Dr. Franco Maria Lajolo/ Dr. Francisco Tomás-Barberán


Antoine Vernay

Philippe Malagoli

Marine Fernandez

Thomas Perot

Thierry Ameglio

Philippe Balandier

Vernay Antoine

Malagoli Philippe

Perot Thomas

Ameglio Thierry

Balandier Philippe

Unexpected impact of N availability on the interaction between Quercus and Deschampsia

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction :

The influence of resource quantity on plant -plant interactions has not led to a consensus. This study aimed to understand how oak tree seedlings and tussock grass interaction evolve among different NL availability combinations.

Objectives :

•

Conclusion:

• When N-fertilised oak seedlings facilitate D. cespitosa growth.

We hypothesize that oaks seedling would produce more exudates available for fast capture by D.cespitosa. • In every cases interaction is negative (competition) for oak.

Higher resource availability (L or N) increased competition.

• The two species display different strategies, capture strategy for grass and conservative strategy for oak. Perspective:

• Integrate other resources as water or phosphorus.

• Assess a potential impact of allelopathic compounds • Determine role of tree seedling reserve to face competition with grass.

•

  Determine how were early oak / D. cespitosa responses affected by abiotic environment • Assess importance and intensity of the interactions, either competition or facilitation • Highlight plant strategy to face interactions with other species Results: More 15 N allocated to oak storage organs, and particularly coarse roots, when mixed with D. cespitosa (Fig 2) • D. cespitosa preferentially allocated 15 N resource to aboveground organs (Fig 2)

Fig 2 :Fig. 1 .

 21 Fig 2: Relative allocation of 15 N among leaves, stems, coarse (CR) and fine roots (FR) in oak seedlings and D. cespitosa (above ground biomass (AG) and below ground biomass (BG)) when sole-(SSp) or mixed-grown (MSp)

PIAF, INRA, Univ. Clermont-Auvergne, 63100 Clermont-Ferrand, France

Irstea, Research Unit on Forest Ecosystems (EFNO), Domaine des Barres, F-45290 Nogent-sur-Vernisson, France


List of Poster presentations

P78

The impact of competitive adsorption between Soy protein and HPMC on emulsion lipolysis, Ana M. R.Pilosof



  To be shown to Tuesday 4 afternoon to Wednesday 5 end of morning

  P1Impact of in vitro digestion on the colloidal stability of powdered flaxseed oil microcapsules, Liliana Gabriela SantiagoP2The microalgae Phaeodactylum tricornutum as a potential carotenoid source for human nutrition, Andrea GilleIn vitro digestion of beta-glucan rich cereal products results in extracts with physiochemical and rheological behavior like pure beta-glucan solutions -A basis for understanding of in vivo effects, Simon Balance P68 Exploring the gastric disintegration of milk protein gels by time-lapse synchrotron deep-UV microscopy, Chenopodium quinoa Willd.) proteins as source of antioxidant and anti-proliferative peptides after in vitro gastrointestinal digestion, Lourdes Amigo P74 Enhancing curcumin bioaccessibility through different nanoformulations, Ana Cristina Pinheiro P75 Potential use of micro-, ultra-and nano-filtration membrane techniques for development of functional ingredients from Theobroma cacao, María de la Luz Cádiz Gurrea P76 Antioxidant and anti-inflammatory activities of bioactive compounds from byproducts, María de la Luz Cádiz Gurrea P77 Different behavior of food polyphenols in energy metabolism of lipopolysaccharide-stimulated cells, María de la Luz Cádiz Gurrea

	P20 P43 P67	Assessment of bioaccessibility and antioxidant potential of flour obtained from cactus pear fruits (Opuntia Impact of droplet triglyceride profiles on emulsion stability and behavior during in vitro pH stat gastrointestinal
		indica) co-products during in vitro gastrointestinal digestion, Juana Fernández López digestion, Dafna Meshulam Pascoviche
	P21 P44	Physical evolution of titanium dioxide particles in digestive fluids, Marie-Hélène Ropers Hydrocolloids modulate the structure breakdown of starch-based systems under oral conditions, A. Agudelo
		Juliane Floury
	P22 P45	Investigating the effect of an inorganic food additive containing nanoparticles on human intestinal microbiota, In vitro study of micellar casein and caseinate digestion, Audrey Boulier
	P69	Marie-Hélène Ropers Healthier biscuits formulated with functional ingredients: in vitro digestion using pH-stat and the INFOGEST
	P46	Study by 1H Nuclear Magnetic Resonance of the evolution of refined soybean oil with different oxidation protocol, Celalettin Degerli
	P23	Plant protein digestibility: The impact of plant cell wall encapsulation and porosity, Mostafa Zahir status under in vitro digestion conditions, Patricia Sopelana
	P70	Effects of excipient foods and food matrix on the carotenoid bioaccessibility, Birgul Hizlar
	P24 P47	The Influence of extraction conditions on in vitro protein digestibility of Spirulina platensis protein extracts: Advance of oxidation during in vitro digestion of edible oils of different unsaturation degree containing
	P71	An optimization study with RSM, Ozcelik Beraat hydroperoxides. A study by SPME-GC/MS and by 1H NMR, Patricia Sopelana In vitro α-amylase inhibition and α-glucosidase inhibition activities of biscuit enriched with anthocyanin and
		docosahexaenoic acid, Sibel Karakaya
	P25 P48	Dried Leafy Green Vegetables : Investigation of In Vitro Bioaccessability of Total Phenolic and Total In vitro gastric digestion of whey protein/hydrocolloids systems. Relationship with satiety capacity, Isabel
	P72	Flavonoid Content, Ozcelik Beraat Hernando Application of the international harmonized in vitro digestion protocol (Infogest) to the digestion of vegetal
		proteins, Lourdes Amigo
	P26 P49	Stability of total phenolic content and antioxidant capacity of a local beetroot (Beta vulgaris L.) variety The Presence of Pectin in Nanoemulsions: Impact on β-carotene Bioaccessibility, Isabel Odriozola-Serrano
	P73	subjected to in vitro digestion, Kadriye Nur Kasapoğlu Quinoa (
	P50	Carotenoids and their esters in pulp of three mandarin cultivars: composition and bioaccessibility, Fabiane
	P27	In vitro protein digestibility of sour cherry kernel (Prunus cerasus L.) protein concentrate, Kadriye Nur C. Petry
		Kasapoğlu
	P51	Oat antioxidant bioactive peptides generation during a simulated human dynamic In Vitro Digestion System
		(IViDiS) model, Yves Arcand
	P52	Impact of varietal differences and gen-environment on protein quality and amino acid digestibility of selected
		Canadian oat varieties: PDCAAS versus DIAAS, Yves Arcand
	P53	Development of an in vitro gastro-intestinal digestion model simulating the young infant pediatric population,
		Anette Müllertz
	P54	Infant's saliva lipolysis: relationships with age, milk and complementary feeding history, Carole Tournier
	P55	Satiety effect of a milk proteins enriched beverage in women after exercise, Nicolina Virgilio
	P56	Galactoglucomannan-derived saccharides as potential prebiotics to counteract the impact of antibiotic
		treatments on the elder gut microbiota and the SCFA production: a preliminary study, Beatriz Míguez
	P57	Xylooligosaccharides from Eucalyptus globulus as promising prebiotics to modulate the elder fecal
		microbiota, Beatriz Míguez
	P58	Food starch structure governs gut microbiome composition, digestion kinetics, and fermentation, Frederick
		Warren
	P59	Interaction of calcium with milk fatty acids during the digestion cheese-type matrices, Erik Ayala Bribiesca
	P36	Bioaccessibility of plant sterols in plant sterol enriched milk-based fruit beverages: comparison of two in vitro
	P13 P60	Novel opioid peptides arising from milk protein gastrointestinal digestion, Isidra Recio methods, Amparo Alegria The effect of β-glukans on the availability of minerals from yogurt, Marek Aljewicz
	P14 P37 P61	Transepithelial transport of lunasin and digestion-derived peptides through Caco-2 monolayers. Inhibitory Paracellular transport of Ara h 6 after endurance exercise, Renata Ariens The impact of baked food matrices on structural and allergenic properties of food allergens, Hadeer Mattar
	P38 P62	effects on gastrointestinal cancer cells proliferation, Isidra Recio Gastric emptying breath test: the importance of the food matrix, Alwine F.M. Kardinaal Multiple emulsions containing Hibiscus Sabdariffa extract stabilized by protein-polysaccharide complexes
	P15	Peptides released during in vitro gastrointestinal digestion from common bean (Phaseolus vulgaris L.). for the development of functional food, Sandra Pimentel-Moral
	P39 P63	Food bolus formation: effect of oral and anthropometric characteristics in function of dental status among Transepithelial transport through Caco-2 cell monolayers, Isidra Recio elderly, Aurelie Prot Pressurized liquid extraction (PLE) for obtaining bioactive ingredients from Hibiscus Sabdariffa, Sandra
	P16	Digestibility of hemp (Cannabis sativa L.) protein isolates, Pasquale Ferranti Pimentel-Moral
	P40	In Situ Characterization of Acidic pH and Cooking Effects on Molecular Structure of Protein by Infrared
	P17 P64	Relating oral processing (OP) and sensory perception of protein-added yogurts, Susana Fiszman Microspectroscopy, Michiyo Motoyama Optimization of extraction of phenolic compounds from a phenolic-enriched functional olive oil, Sandra
		Pimentel-Moral
	P18 P41	Analysis of antioxidant capacity and bioaccessibility of buckwheat enriched bakery products in model Static vs dynamic in vitro digestions of an innovative Citrus concentrate: Bioaccessibility of its
	P65	simulating gastrointestinal human tract, Małgorzata Wronkowska phytomicronutrients, Laura Gence In vitro starch hydrolysis of common bean process-induced structures, Tara Grauwet
	P19 P42 P66	Bioaccessibility and antioxidant potential of flour obtained from persimmon fruit (Diospyros kaki) co-products Caco-2 cells: a novel identification tool for DPP-IV inhibitors, Benoit Cudennec In vitro lipolysis and carotenoid bioaccessibility of emulsified oils with different unsaturation degree: a kinetic
		during in vitro gastrointestinal digestion, Juana Fernández López study, Tara Grauwet

P3 Digestion of Proteins in Milk: Comparing different in vitro systems with in vivo data, Lotti Egger P4 Protein carriers for folic acid: characterization of the interaction and bioaccessibility of the vitamin after in vitro digestion, Carolina Arzeni P5 Fine fruits enriched pasta: in vitro antioxidant activity and starch digestibility, Mariela Cecilia Bustos Shmidt P6 Sorghum gluten free pasta: In vitro digestion and dialyzable antioxidant activity, Mariela Cecilia Bustos Shmidt P7 Nutritional Quality and glycemic index evaluation of whole grain flours and breads, Mariela Cecilia Bustos Shmidt P8 Study on stability of vitamin B12/whey protein complexes and their effects on gut microbiome, Xuan Zhu P9 Pepsin localization in digested meat: a Tof-SIMS analysis, Thierry Astruc P10 In vivo MRI studies of blood glucose and gastrointestinal responses to breakfast porridge made from modern and ancient grains, Jaber Alyami P11 Beta-glucan and its effect on lipid metabolism, Myriam Grundy P12 In vitro protein digestion in the allergenicity assessment of (novel) proteins, Fernandez Dumont Antonio P28 Kinetics and mechanisms of N-nitrosation and nitrosylation in a gastro-intestinal model, Philippe Gatellier P29 A first step towards a consensus static in vitro model for simulating full-term newborn digestion: application to infant formulas, Olivia Menard P30 Lipid oxidation and its inhibition by apple polyphenols in static and dynamic in vitro digestion systems, Claire Dufour P31 Study of in vitro bioaccessibility of vitamin A naturally present in cod liver oil by means of 1H NMR and SPME-GC/MS, Barbara Nieva-Echevarría P32 Antioxidant activity of ovalbumin during in vitro gastrointestinal digestion of oils rich in ω-6 and ω-3 lipids. A study by 1H NMR and SPME-GC/MS, Barbara Nieva-Echevarría P33 Rearing in hot environments reduces the transepithelial resistance of Ross 308 broilers, Frank Dunshea P34 Bioconversion by gut microbiota of predigested mango (Mangifera indica L) peel polyphenols assessed in a dynamic in vitro model of the human colon?, Sonia G Sayago-Ayerdi.

P35

In vitro colonic fermentation of Mexican "taco" from corn tortilla and black beans in a Simulator of Human Microbial Ecosystem (SHIME®) system? Sonia G Sayago-Ayerdi.


Keywords: Biochemistry, Physiology, and Clinical Applications

teaching and research institutions in France or abroad, or from public or private research centers.

K22989

Glutamine: Biochemistry, Physiology, and Clinical Applications describes the different functions of glutamine (Gln) in animals and humans. Gln is both a nutrient and a signaling molecule, and its functions go beyond those of a simple metabolic fuel or protein precursor. This book has gathered together, in an unbiased and critical manner, all the available evidence and research on Gln including pathology (neurological diseases, intestinal diseases, critical illness, and cancer), physiology (successful aging), catabolic states, immunity, and exercise. Special attention is given to the potential bene t of Gln in states of insulin resistance and the role of Gln as a "conditionally essential" amino acid.

Features

• Promotes the signaling role of Gln in autophagy, which is an emerging concept in aging due to the fact that its increase can favor longevity. • Considers how a high concentration of Gln and its accumulation in plasma and in the brain can induce toxicity and lead to severe central nervous system diseases. • Demonstrates how Gln supplementation is bene cial for the recovery of intestinal mucosal integrity and for intestinal diseases and aging. • Discusses why Gln supplementation in catabolic states is only warranted in the event of hypoglutaminemia and the absence of renal or hepatic failure. • Highlights how Gln, as a promising biomarker or even as a therapeutic target in cancer, could have an important role in effective anticancer strategies. • Includes future directions of Gln research.

The contributors are either pioneers or experts in the area of Gln from all around the globe, including Australia, Brazil, Canada, China, Europe, and the United States. This book is a valuable source of information for nutrition scientists, medical doctors, sports scientists, food scientists, dietitians, and anyone interested in nutrition. It is also a valuable resource for students in these elds and will be an important addition to university libraries.

Glutamine Biochemistry, Physiology, and Clinical Applications

  T A M I N E Meynial-Denis


Daniele A Di Pietro

Alexandre Ern

DISCRETE FUNCTIONAL ANALYSIS TOOLS FOR DISCONTINUOUS GALERKIN METHODS WITH APPLICATION TO THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier-Stokes equations. Two discrete convective trilinear forms are proposed, a non-conservative one relying on Temam's device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure.

Introduction

Discontinuous Galerkin (DG) methods were introduced over thirty years ago to approximate hyperbolic and elliptic PDE's (see e.g. [START_REF] Arnold | Unified analysis of discontinuous Galerkin methods for elliptic problems[END_REF][START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. I. General theory[END_REF] for a historical perspective), and they have received extensive attention over the last decade. For linear PDE's, the mathematical analysis of such methods is well-understood; see e.g. [START_REF] Arnold | Unified analysis of discontinuous Galerkin methods for elliptic problems[END_REF] for a unified analysis for the Poisson problem, [START_REF] Di Pietro | Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection[END_REF] for advection-diffusion equations with semidefinite diffusion, and [START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. I. General theory[END_REF][START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order elliptic PDEs[END_REF][START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. III. Multi-field theories with partial coercivity[END_REF] for a unified analysis encompassing hyperbolic and elliptic PDE's in the framework of Friedrichs' systems. The situation is substantially different when dealing with nonlinear second-order PDE's. Indeed, although DG methods have been widely used for such problems, their mathematical analysis has hinged almost exclusively on strong regularity assumptions on the exact solution. This is in stark contrast with the recent literature on Finite Volume (FV) schemes where, following the penetrating works of Eymard, Gallouët, Herbin and co-authors (see e.g. [START_REF] Eymard | The Finite Volume Method[END_REF][START_REF] Eymard | Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes[END_REF][START_REF] Eymard | Analysis tools for finite volume schemes[END_REF]), new discrete functional analysis tools have been derived allowing to prove the convergence to minimum regularity solutions, i.e. solutions belonging to the natural function spaces in which the weak formulation of the PDE is set. The key ideas can be summarized as follows:

(i) an a priori estimate on the discrete solution and an associated compactness result are used to infer the strong convergence of a subsequence of discrete solutions to a function u in some Lebesgue space, say L 2 (Ω); (ii) the construction of a discrete gradient converging to ∇u in a suitable Lebesgue space allows to prove that the limit u actually belongs to some space with additional regularity, say H 1 0 (Ω);

Date: Draft version: May 14, 2008.

(iii) the convergence of the scheme is finally proved testing against a discrete projection of a smooth function belonging to some convenient dense subspace, say C ∞ c (Ω). When the exact solution is unique, the convergence of the whole sequence of discrete approximations is deduced. Moreover, stronger convergence results on the discrete gradient can be derived using the dissipative structure of the problem at hand whenever available. In the present work we show how the analysis tools derived for FV schemes can be extended to DG methods using the steady incompressible Navier-Stokes equations as a model problem. Discontinuous Galerkin approximations of the steady incompressible Navier-Stokes problem have been derived in recent works using different techniques; see, among others, [START_REF] Bassi | An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations[END_REF][START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF][START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF][START_REF] Karakashian | A nonconforming finite element method for the stationary Navier-Stokes equations[END_REF][START_REF] Mozolevski | Discontinuous Galerkin finite element approximation of the two-dimensional Navier-Stokes equations in stream-function formulation[END_REF].

The present analysis relies on two discrete functional analysis tools in piecewise polynomial spaces on general meshes of a bounded Lipschitz domain Ω ⊂ R d (DG spaces henceforth). Firstly, upon introducing the usual • DG -norm consisting of the broken gradient plus a jump term (see [START_REF] Brenner | Poincaré-Friedrichs inequalities for piecewise H 1 functions[END_REF]) as well as non-Hilbertian variants thereof (see §6), we prove discrete Sobolev embeddings that are the counterpart of those valid at the continuous level, v L q (Ω) ≤ S p,q ∇v L p (Ω) d , ∀v ∈ W 1,p 0 (Ω), for suitable indices q and p. Probably the best known discrete embedding of such a type is the so-called broken Poincaré-Friedrichs inequality obtained with p = q = 2; see e.g. [START_REF] Arnold | An interior penalty finite element method with discontinuous elements[END_REF][START_REF] Brenner | Poincaré-Friedrichs inequalities for piecewise H 1 functions[END_REF]. The broken Sobolev embeddings we are concerned with have been derived by Lasis and Süli [START_REF] Lasis | Poincaré-type inequalities for broken Sobolev spaces[END_REF] in the Hilbertian case (p = 2). Those we establish in a non-Hilbertian setting (p = 2) are, to the best of our knowledge, new. The proofs are substantially different from the ones used in the finite element literature, which rely on elliptic regularity and or on nonconforming finite element interpolants. Indeed, we take inspiration from the techniques used in [START_REF] Eymard | Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes[END_REF] in the case of piecewise constant functions. A crucial point is the observation that the BV norm defined in Lemma 6.2 hereafter is controlled by the • DG -norm and also by non-Hilbertian variants thereof. The present technique of proof readily incorporates the use of general meshes, an important feature when working with DG methods. Observe, however, that we only establish the embedding results in DG spaces only, and not in the larger setting of broken Sobolev spaces. The latter are indeed not used in the convergence proofs below.

The second functional analysis tool, which is, to the best of our knowledge, new in the framework of DG methods, is a compactness result for bounded sequences in the • DG -norm and non-Hilbertian versions thereof. Here again, the proof is quite simple and it is inspired from [START_REF] Eymard | Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes[END_REF]: it consists of using Kolmogorov's Compactness Criterion (see e.g. [START_REF] Brezis | Analyse fonctionnelle[END_REF]Theorem IV.25]) based on uniform translates estimates in L 1 (R d ) together with the above discrete Sobolev embeddings and a discrete gradient operator that is shown to be weakly convergent in some L p (Ω) space with p > 1.

This paper is organized as follows. §2 introduces the discrete setting, including the assumptions on the meshes, the DG spaces, and the discrete gradient operators, whose weak convergence is proven in Theorem 2.2. §3 is concerned with the Poisson problem; its purpose is to show how the diffusive term is analyzed. The main result is Theorem 3.1. §4 deals with the Stokes equations; its purpose is to show how the velocity-pressure coupling is handled. The main result is Theorem 4.1. §5 is concerned with the steady incompressible Navier-Stokes equations; its main result is Theorem 5.1. Two discrete convective trilinear forms are proposed, a non-conservative one relying on Temam's device to control the kinetic energy balance [START_REF] Temam | Theory and numerical analysis[END_REF] and a conservative one based on a nonstandard modification of the pressure hinted to in [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF]. Finally, §6 contains the discrete functional analysis tools in DG spaces. The main results are Theorem 6.1 and 6.3 which are presented in a non-Hilbertian setting since their validity extends beyond the model problems considered in this work.

T ′ ∈ T h such that F = ∂T ∩ ∂T ′ ; (iii) there is a parameter ̺ 1 independent of h such that for all T ∈ T h , (1)
F ⊂∂T h F |F | ≤ ̺ 1 |T |,
where h F denotes the diameter of the face F , |F | its (d -1)-dimensional measure and |T | the d-dimensional measure of T ; (iv) for all h ∈ H, each T ∈ T h is affine-equivalent to an element belonging to a finite collection of reference elements; (v) the ratio of the diameter h T of any T ∈ T h to the diameter of the largest ball inscribed in T is bounded from above by a parameter ̺ 2 independent of h; (vi) there is a parameter ̺ 3 , independent of h, such that for all T ∈ T h and for all faces

F ⊂ ∂T , h F |F | ≥ ̺ 3 |T |.
For each h ∈ H, we define size(T h ) def = max T ∈T h h T . The parameters introduced in the above definition will be referred to as the basic mesh parameters and collectively denoted by the symbol P.

Remark 2.1. Assumption (vi) is needed only when working with a particular choice of the stabilization bilinear form penalizing interelement jumps. It can be lifted by working with other forms; see Remark 3.2 for further discussion. Furthermore, assumption (v) will not be needed in §6 to prove the discrete Sobolev embeddings nor the weak convergence of discrete gradients. Figure 1 presents an example of admissible mesh in two space dimensions. The mesh faces are collected in the set F h . The set F h will be partitioned into F i h ∪ F b h , where F b h collects the faces located on the boundary of Ω and F i h the remaining ones. For F ∈ F i h , there are T 1 and T 2 in T h such that F = ∂T 1 ∩ ∂T 2 , and we define ν F as the unit normal vector to F pointing from T 1 to T 2 . For any function Figure 1. An example of admissible mesh ϕ such that a (possibly two-valued) trace is defined on F , let ( 2)

ϕ def = ϕ |T1 -ϕ |T2 , { {ϕ} } def = 1 2 (ϕ |T1 + ϕ |T2 ).
For F ∈ F b h , ν F is defined as the unit outward normal to Ω, while the jump and average are conventionally defined as

ϕ def = ϕ and { {ϕ} } def = ϕ.
For any integer k ≥ 0 and for all T ∈ T h , let P k (T ) denote the vector space of polynomial functions defined on T with real coefficients and with total degree less than or equal to k. Owing to assumptions (iii) and (iv) in Definition 2.1, there is c k,P such that, for all h ∈ H and for all T ∈ T h , [START_REF] Bassi | An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations[END_REF] ∀v h ∈ P k (T ),

F ⊂∂T h F F |v h | 2 ≤ c k,P T |v h | 2 .
Here and in what follows, the symbol c will be used to denote a positive generic constant whose value can change at each occurrence. To keep track of the dependency of such constants on some parameters, subscripts will be used whenever relevant.

2.2. DG spaces. Let k ≥ 0 and consider the finite dimensional space (4)

V k h def = {v h ∈ L 2 (Ω); ∀T ∈ T h , v h|T ∈ P k (T )}.
For k ≥ 1, this space is equipped with the norm

(5) v h 2 DG def = T ∈T h T |∇v h | 2 + F ∈F h 1 h F F | v h | 2 ,
where |•| denotes the Euclidean norm in R d . For further use, it will be convenient to introduce the seminorm

(6) |v h | 2 J,F ,±1 def = F ∈F h ±1 F F | v h | 2 ,
where F is a subset of F h that will usually be taken equal to F h or to

F i h . Moreover, we define ∇ h v h as the piecewise gradient of v h ∈ V k h , i.e., ∇ h v h ∈ [V k-1 h ] d is such that for all T ∈ T h , ∇ h v h|T = ∇(v h|T ), so that (7) v h 2 DG = ∇ h v h 2 L 2 (Ω) d + |v h | 2 J,F h ,-1 .
The above norm and seminorm can be extended to

H 1 + (Ω) def = H 1 0 (Ω)+V k h (actually, an extension to C ∞ c (Ω) + V k
h is sufficient for the present purposes). A straightforward but important result concerns the approximability of smooth functions in the • DG -norm. For all l ≥ 0, let π l h denote the

L 2 (Ω)-orthogonal projection from L 2 (Ω) onto V l h . Let ϕ ∈ C ∞ c (Ω).
Then, owing to assumptions (iii)-(v) in Definition 2.1, it is clear using classical approximation properties (see e.g. [START_REF] Brenner | The Mathematical Theory of Finite Element Methods[END_REF][START_REF] Ern | Theory and Practice of Finite Elements[END_REF]) that for all l ≥ 1, [START_REF] Brezzi | Discontinuous Galerkin approximations for elliptic problems[END_REF] ϕ -π l h ϕ DG → 0 as size(T h ) → 0. In what follows, we shall make frequent use of the projector π 1 h which will be simply denoted by π h . For l = 0 we shall use the following property: [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF] ϕ -π 0 h ϕ L ∞ (Ω) → 0 as size(T h ) → 0. The above projectors will also be applied componentwise to vector-valued functions. The following stability result holds: For all v ∈ H 1 (Ω), ( 10)

∇ h π k h v 2 L 2 (Ω) d + F ∈F h h -1 F v -π k h v 2 L 2 (F ) ≤ c k,P v 2 H 1 (Ω) .
For ease of exposition, we restate hereafter the consequences of Theorem 6.1 in the Hilbertian setting. The proof will be given in §6.

Theorem 2.1 (Discrete Sobolev embeddings). For all q such that (i

) 1 ≤ q ≤ 2d d-2 if d ≥ 3; (ii) 1 ≤ q < +∞ if d = 2; there is σ q such that (11) ∀v h ∈ V k h , v h L q (Ω) ≤ σ q v h DG .
The constant σ q additionally depends on k, |Ω|, and P.

2.3. Discrete gradient operators. For all F ∈ F h , let r l F : L 2 (F ) → [V l h ] d , l ≥ 0, be the lifting operator defined as follows: For all φ ∈ L 2 (F ), [START_REF] Deimling | Nonlinear functional analysis[END_REF]

∀τ h ∈ [V l h ] d , Ω r l F (φ)•τ h = F { {τ h } }•ν F φ.
Clearly, the support of r l F (φ) consists of the one or two mesh elements of which F is a face. For v ∈ H 1 + (Ω), define

(13) R l h (v) def = F ∈F h r l F (v). Let now k ≥ 1. The following discrete gradient operators G l h : V k h → [V max(k-1,l) h
] d will play an important role in the subsequent analysis [START_REF] Di Pietro | Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection[END_REF]

∀v h ∈ V k h , G l h (v h ) def = ∇ h v h -R l h ( v h ) = ∇ h v h - F ∈F h r l F ( v h ).
For a given k ≥ 1, the most natural value for l is k or (k -1), but the values l = 0 and l = 2k will also be used.

∀v h ∈ V k h , G l h (v h ) L 2 (Ω) d ≤ c k,l,P v h DG . (15)
Proof. It is straightforward to verify using assumption (ii) in Definition 2.1 that for all

v h ∈ V k h , (16) R l h ( v h ) 2 L 2 (Ω) d ≤ N ∂ F ∈F h r l F ( v h ) 2 L 2 (Ω) d .
Furthermore, owing to the trace inequality (3) and proceeding as in [START_REF] Brezzi | Discontinuous Galerkin approximations for elliptic problems[END_REF], it is inferred that for all F ∈ F h , (17)

r l F ( v h ) 2 L 2 (Ω) d ≤ c k,l,P 1 h F F | v h | 2 .
As a result,

R l h ( v h ) 2 L 2 (Ω) d ≤ c k,l,P |v h | 2 J,F h ,-1 .
Using the triangle inequality yields [START_REF] Ern | Theory and Practice of Finite Elements[END_REF].

The main property of the discrete gradient operators defined by ( 14) is their weak convergence in L 2 (Ω) d when evaluated on bounded sequences in the • DG -norm. Theorem 2.2 (Compactness and weak convergence of discrete gradients). Let k ≥ 1 and let l ≥ 0. Let {v h } h∈H be a sequence in V k h . Assume that this sequence is bounded in the • DG -norm. Then, there exists a function v ∈ H 1 0 (Ω) such that as size(T h ) → 0, up to a subsequence, v h → v strongly in L 2 (Ω) and for all l ≥ 0, G l h (v h ) ⇀ ∇v weakly in L 2 (Ω) d . Proof. Owing to Theorem 6.2 applied with p = 2 and extending the functions v h by zero outside Ω, there exists a subsequence still denoted {v h } h∈H and a function

v ∈ L 2 (R d ) such that as size(T h ) → 0, v h → v strongly in L 2 (R d ). Moreover, since {G l h (v h )} h∈H is bounded in L 2 (R d ) d owing to Proposition 2.1, up to a new subsequence, there is w ∈ L 2 (R d ) d s.t. G l h (v h ) ⇀ w weakly in L 2 (R d ) d . To prove that w = ∇v, let ϕ ∈ C ∞ c (R d ) d and observe that R d G l h (v h )•ϕ = - R d v h (∇•ϕ) - R d R l h ( v h )•(ϕ -π 0 h ϕ) + F ∈F h F { {ϕ -π 0 h ϕ} }•ν F v h = T 1 + T 2 + T 3 .
Letting size(T h ) → 0, we observe that

T 1 → -R d v(∇•ϕ) and that T 2 → 0 since ϕ -π 0 h ϕ L ∞ (R d ) d → 0 and {R l h ( v h )} h∈H is bounded in L 2 (R d ) d
. Furthermore, the Cauchy-Schwarz inequality, together with assumption (iii) in Definition 2.1, yields

|T 3 | ≤ C ϕ -π 0 h ϕ L ∞ (R d ) d |v h | J,F h ,-1 ≤ C ′ ϕ -π 0 h ϕ L ∞ (R d ) d
, which tends to zero as size(T h ) → 0 owing to [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF]. As a result,

R d w•ϕ = lim size(T h )→0 R d G l h (v h )•ϕ = - R d v(∇•ϕ), implying that w = ∇v. Hence, v ∈ H 1 (R d ) and since v is zero outside Ω, v is in H 1 0 (Ω).
It is useful to introduce for all l ≥ 0, further discrete gradient operators G l h :

V k h → [V max(k-1,l) h ] d s.t. ( 18
) ∀v h ∈ V k h , G l h (v h ) def = ∇ h v h - F ∈F i h r l F ( v h ).
The difference with the discrete gradient operator G l h defined by ( 14) is that boundary faces are not included in [START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. III. Multi-field theories with partial coercivity[END_REF]. Clearly, the discrete gradient operators G l h also satisfy the stability property [START_REF] Ern | Theory and Practice of Finite Elements[END_REF]. More importantly, these operators also satisfy the conclusions of Theorem 2.2. This is so because ϕ in the above proof is compactly supported; hence, as size(T h ) → 0, the mesh becomes fine enough so that all the mesh elements having a boundary face are located outside the support of ϕ.

The Poisson problem

Let f ∈ L r (Ω) with r = 2d d+2 if d ≥ 3 and r > 1 if d = 2. Set r ′ def = r r-1 . Consider the following model problem (19) -∆u = f, in Ω, u = 0, on ∂Ω.
The weak formulation of this problem consists of finding u ∈

H 1 0 (Ω) s.t. for all v ∈ H 1 0 (Ω), (20)
Ω ∇u•∇v = Ω f v.
It is well-known that this problem is well-posed. In particular, owing to the Sobolev

embedding v L r ′ (Ω) ≤ S 2,r ′ ∇v L 2 (Ω) d valid for all v ∈ H 1 0 (Ω)
, and using Hölder's inequality, it is inferred that

(21) ∇u 2 L 2 (Ω) d = Ω f u ≤ f L r (Ω) u L r ′ (Ω) ≤ S 2,r ′ f L r (Ω) ∇u L 2 (Ω) d , yielding the a priori bound ∇u L 2 (Ω) d ≤ S 2,r ′ f L r (Ω) . 3.1. Symmetric formulation. Let k ≥ 1.
For the sake of simplicity, discrete gradients are built using the lifting operators r k F (see Remark 3.2 for further discussion) and to alleviate the notation, the superscript k is omitted. This convention is kept for the rest of this work. For all (v h , w h ) ∈ V k h × V k h , consider the following symmetric DG bilinear form [START_REF] Eymard | Analysis tools for finite volume schemes[END_REF] a

h (v h , w h ) def = Ω G h (v h )•G h (w h ) + j h (v h , w h ),
with the stabilization bilinear form

(23) j h (v h , w h ) def = F ∈F h η Ω r F ( v h )•r F ( w h ) - Ω R h ( v h )•R h ( w h ),
where η ∈ R + is a penalty parameter. Henceforth, we assume that

(24) η > N ∂ . Lemma 3.1 (Continuity). For all (v, w) ∈ H 1 + (Ω) × H 1 + (Ω), (25) a h (v, w) ≤ c η,k,P v DG w DG . Proof. For all (v, w) ∈ H 1 + (Ω) × H 1 + (Ω), the Cauchy-Schwarz inequality yields a h (v, w) 2 ≤ ( G h (v) 2 L 2 (Ω) d + j h (v, v))( G h (w) 2 L 2 (Ω) d + j h (w, w)). Clearly, for v ∈ H 1 + (Ω) such that v = v 1 + v h with v 1 ∈ H 1 0 (Ω) and v h ∈ V k h , G h (v) L 2 (Ω) d ≤ ∇ h v L 2 (Ω) d + R h (v h ) L 2 (Ω) d since R h (v 1 ) = 0. Proposition 2.1 then yields R h (v h ) L 2 (Ω) d ≤ c k,P |v h | J,F h ,-1 = c k,P |v| J,F h ,-1 since |v 1 | J,F h ,-1 = 0. As a result, G h (v) L 2 (Ω) d ≤ c k,P v DG ,
and similarly,

j h (v, v) ≤ (N ∂ + η)c k,P |v| 2 J,F h ,-1 , completing the proof. Lemma 3.2 (Coercivity). For all v h ∈ V k h , (26)
G h (v h ) 2 L 2 (Ω) d + (η -N ∂ ) F ∈F h r F ( v h ) 2 L 2 (Ω) d ≤ a h (v h , v h ).
Furthermore, there is α > 0, depending on η, k and P such that for all 26) directly results from [START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. I. General theory[END_REF]. To verify [START_REF] Kovasznay | Laminar flow behind a two-dimensional grid[END_REF], observe first that proceeding as in [START_REF] Brezzi | Discontinuous Galerkin approximations for elliptic problems[END_REF] using assumptions (iv) and (vi) in Definition 2.1 yields for all

v h ∈ V k h , (27) α v h 2 DG ≤ a h (v h , v h ). Proof. Estimate (
F ∈ F h , (28) 1 h F F | v h | 2 ≤ c ′ k,P r F ( v h ) 2 L 2 (Ω) d .
Using the triangle inequality, it is then inferred that

v h 2 DG ≤ 2 G h (v h ) 2 L 2 (Ω) d + 2 R h ( v h ) 2 L 2 (Ω) d + |v h | 2 J,F h ,-1 ≤ 2 G h (v h ) 2 L 2 (Ω) d + (2N ∂ + (c ′ k,P ) -1 ) F ∈F h r F ( v h ) 2 L 2 (Ω) d ≤ max(2, (2N ∂ + (c ′ k,P ) -1 )(η -N ∂ ) -1 )a h (v h , v h
), the last inequality resulting from [START_REF] Karakashian | A nonconforming finite element method for the stationary Navier-Stokes equations[END_REF].

Remark 3.1. A straightforward calculation shows that

a h (v h , w h ) = Ω ∇ h v h •∇ h w h - F ∈F h F ν F •{ {∇ h v h } } w h + ν F •{ {∇ h w h } } v h + F ∈F h η Ω r F ( v h )•r F ( w h ),
yielding the IP-type method introduced in [START_REF] Bassi | A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows[END_REF]. It is also possible to consider the stabilization bilinear form

j SIPG h (v h , w h ) def = F ∈F h η 1 h F F v h w h - Ω R h ( v h )•R h ( w h ),
yielding the usual Symmetric Interior Penalty method (SIPG) [START_REF] Arnold | An interior penalty finite element method with discontinuous elements[END_REF]. In this case, the minimal threshold for the penalty parameter η depends on the constant in the trace inequality [START_REF] Bassi | An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations[END_REF]. It is also possible to consider the stabilization bilinear form

j LDG h (v h , w h ) def = F ∈F h η 1 h F F v h w h ,
yielding the usual Local Discontinuous Galerkin method (LDG) [START_REF] Cockburn | The local discontinuous Galerkin finite element method for convection-diffusion systems[END_REF]. The advantage is that the parameter η needs only be positive, the disadvantage is however that the stencil is enlarged to neighbors of neighbors. Moreover, working with any of the two above stabilization bilinear forms allows to lift assumption (vi) in Definition 2.1.

For all h ∈ H, Lemma 3.2 implies that there is a unique

u h ∈ V k h s.t. ( 29
) a h (u h , v h ) = Ω f v h , ∀v h ∈ V k h .
Theorem 3.1 (Convergence for Poisson problem). Let {u h } h∈H be the sequence of approximate solutions generated by solving the discrete problems (29) on the admissible meshes {T h } h∈H . Then, as size(T h ) → 0,

u h → u, in L 2 (Ω), (30)
∇ h u h → ∇u, in L 2 (Ω) d , ( 31
)
|u h | J,F h ,-1 → 0, ( 32
)
where u ∈ H 1 0 (Ω) is the unique solution to [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF]. Proof. (i) A priori estimate. Using Lemma 3.2 and Hölder's inequality, it is inferred that

α u h 2 DG ≤ a(u h , u h ) = Ω f u h ≤ f L r (Ω) u h L r ′ (Ω) .
Hence, owing to Theorem 2.1, the sequence {u h } h∈H is bounded in the • DG -norm. (ii) L 2 -convergence of a subsequence, regularity of the limit and weak convergence of discrete gradient. Owing to Theorem 2.2, there exists u ∈ H 1 0 (Ω) such that, as size(T h ) → 0, up to a subsequence,

u h → u strongly in L 2 (Ω) and G h (u h ) ⇀ ∇u weakly in L 2 (Ω) d . (iii) Identification of u and convergence of the whole sequence. Let us first prove that for all ϕ ∈ C ∞ c (Ω), (33) a h (u h , π h ϕ) → Ω ∇u•∇ϕ.
Indeed, observe that

a h (u h , π h ϕ) = a h (u h , π h ϕ -ϕ) + Ω G h (u h )•∇ϕ = T 1 + T 2 .
Clearly, T 1 → 0 owing to Lemma 3.1 since u h DG is bounded and ϕ -π h ϕ DG converges to zero. Furthermore, T 2 → Ω ∇u•∇ϕ owing to the weak convergence of the discrete gradient. A direct consequence of (33

) is that for all ϕ ∈ C ∞ c (Ω), Ω f ϕ ← Ω f π h ϕ = a h (u h , π h ϕ) → Ω ∇u•∇ϕ.
Thus, u solves the Poisson problem by density of C ∞ c (Ω) in H 1 0 (Ω). Since the solution to this problem is unique, the whole sequence {u h } h∈H strongly converges to u in L 2 (Ω) and {G h (u h )} h∈H weakly converges to ∇u in L 2 (Ω) d . (iv) Strong convergence of the discrete gradient and of the jumps. Owing to [START_REF] Karakashian | A nonconforming finite element method for the stationary Navier-Stokes equations[END_REF] and to weak convergence,

lim inf a h (u h , u h ) ≥ lim inf G h (u h ) 2 L 2 (Ω) d ≥ ∇u 2 L 2 (Ω) d . Furthermore, still owing to (26), G h (u h ) 2 L 2 (Ω) d ≤ a h (u h , u h ) = Ω f u h , yielding lim sup G h (u h ) 2 L 2 (Ω) d ≤ lim sup a h (u h , u h ) = lim sup Ω f u h = Ω f u = ∇u 2 L 2 (Ω) d . Thus, G h (u h ) L 2 (Ω) d → ∇u L 2 (Ω) d , classically yielding the strong convergence of the discrete gradient in L 2 (Ω) d . Note that a h (u h , u h ) → ∇u 2 L 2 (Ω) d also. Finally, owing to (26), (η -N ∂ ) F ∈F h r F ( u h ) 2 L 2 (Ω) d ≤ a h (u h , u h ) -G h (u h ) 2 L 2 (Ω) d ,
and since η > N ∂ and the right-hand side tends to zero, it is inferred using ( 28) that |u h | J,F h ,-1 → 0. Moreover, using [START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. I. General theory[END_REF] to estimate the second term yields

∇ h u h -∇u L 2 (Ω) d ≤ G h (u h ) -∇u L 2 (Ω) d + R h (u h ) L 2 (Ω) d → 0,
as size(T h ) → 0, concluding the proof.

Remark 3.2. To prove the convergence of the method, it is sufficient to work with the lifting operators r 0 F . However, if the exact solution u turns out to be more regular, optimal-order convergence rates can be established in the • DG -norm when working with the lifting operators r k-1 F or r k F . (The latter choice may be preferable for implementation purposes, especially if non-hierarchical, e.g. nodal-based, basis functions are used.) For instance, if u belongs to the broken Sobolev space H k+1 (T h ), the usual a priori error analysis techniques can be used to infer a bound of the form u -u h DG ≤ c u size(T h ) k . A minor difference with the somewhat more usual formulation which does not employ explicitly the discrete gradient operators, is that (29) is only weakly consistent, but not strongly consistent. Indeed, it is easily seen that for all

v h ∈ V k h , a h (u h -u, v h ) = F ∈F h F ν F •{ {π k h (∇u) -∇u} } v h ≤ c u size(T h ) k v h DG . 3
a h (v h , w h ) = Ω G h (v h )•G h (w h ) + j ′ h (v h , w h ),
where G h and G h are discrete gradient operators and where the stabilization bilinear form j ′ h can differ from that given by [START_REF] Eymard | Convergence analysis of a colocated finite volume scheme for the incompressible navier-stokes equations on general 2D or 3D meshes[END_REF]. The following design conditions must be satisfied.

(i) Control on discrete gradients: there is c s.

t. for all v ∈ H 1 + (Ω), (35) G h (v) L 2 (Ω) d + G h (v) L 2 (Ω) d ≤ c v DG ; (ii) Strong consistency of the discrete gradient G h for smooth functions: for all ϕ ∈ C ∞ c (Ω), G h (ϕ) = ∇ϕ;
(iii) Weak consistency of the discrete gradient G h : for any sequence

{v h } h∈H converging in L 2 (Ω) to a function v ∈ H 1 0 (Ω) and such that G h (v h ) L 2 (Ω) d is bounded, up to a subsequence, G h (v h ) ⇀ ∇v weakly in L 2 (Ω) d ; (iv) Stabilization: the bilinear form j ′ h is symmetric and positive, there is c s.t. for all v ∈ H 1 + (Ω), j h (v, v) ≤ c|v| 2 J,F h ,-1 and there is η * > 0 such that for all v h ∈ V k h , (36) a h (v h , v h ) ≥ η * v h 2 DG .
Observe that the continuity property of j h implies that for all ϕ ∈ C ∞ c (Ω), j h (ϕ, •) = 0 and that (36) implies that the discrete problem (34) is well-posed. Under the above assumptions, the convergence of the sequence of discrete DG approximations can be proven. The proof, however, proceeds along a slightly different path with respect to the symmetric formulation. Theorem 3.2. Let {u h } h∈H be the sequence of approximate solutions generated by solving the discrete problems [START_REF] Lew | Optimal BV estimates for a discontinuous Galerkin method for linear elasticity[END_REF] with the bilinear form a h given by (34) on the admissible meshes {T h } h∈H . Assume the above design conditions (i)-(iv). Then, as size(T h ) → 0,

u h → u in L 2 (Ω) and G h (u h ) → ∇u in L 2 (Ω) d .
Proof. (i) Proceeding as before, it is inferred from (iv) that the sequence {u h } h∈H is bounded in the • DG -norm so that there exists u ∈ H 1 0 (Ω) such that, up to a subsequence,

u h → u in L 2 (Ω) as size(T h ) → 0. Furthermore, since the sequence {G h (u h )} h∈H is bounded in L 2 (Ω) d owing to (i), the weak consistency property (iii) yields that (up to a new subsequence) G h (u h ) weakly converges to ∇u in L 2 (Ω) d . (ii) Strong convergence of G h (u h ) ∈ L 2 (Ω) d . Let ϕ ∈ C ∞ c (Ω). Observe that 1 2 G h (u h ) -∇u 2 L 2 (Ω) d ≤ G h (u h ) -G h (π h ϕ) 2 L 2 (Ω) d + G h (π h ϕ) -∇u 2 L 2 (Ω) d = T 1 + T 2 .
Clearly, as size(T h ) → 0,

T 2 → ∇(ϕ-u) 2 L 2 (Ω) d since, using (i) and (ii), G h (π h ϕ)- ∇ϕ = G h (π h ϕ -ϕ) is bounded in L 2 (Ω) d by ϕ -π h ϕ DG which converges to zero.
To bound T 1 , use (i) and (iv) to infer

T 1 = G h (u h -π h ϕ) 2 L 2 (Ω) d ≤ c η * a h (u h -π h ϕ, u h -π h ϕ) = c η * Ω f (u h -π h ϕ) -a h (π h ϕ, u h -π h ϕ) = c η * (T 1,1 -T 1,2 ). Clearly, as size(T h ) → 0, T 1,1 → Ω f (u -ϕ). Moreover, by definition, T 1,2 = Ω G h (π h ϕ)•G h (u h -π h ϕ) + j ′ h (π h ϕ, u h -π h ϕ). Since G h (π h ϕ) strongly converges to ∇ϕ in L 2 (Ω) d and G h (u h -π h ϕ) weakly converges to ∇(u -ϕ) in L 2 (Ω) d
, the first term in the right-hand side converges to

Ω ∇ϕ•∇(u -ϕ).
In addition, the second term is equal to j ′ h (π h ϕ -ϕ, u h -π h ϕ) which converges to zero. Collecting the above bounds, it is inferred that

lim sup G h (u h ) -∇u 2 L 2 (Ω) d ≤ C u -ϕ 2 H 1 (Ω) . Letting ϕ ∈ C ∞ c (Ω) tend to u in H 1 0 (Ω)
, the upper bound can be made as small as desired. This implies the strong convergence of G h (u h ) to ∇u in L 2 (Ω) d .

(iv) Identification of the limit and convergence of the whole sequence.

Let ϕ ∈ C ∞ c (Ω). It is clear that as size(T h ) → 0, Ω f π h ϕ → Ω f ϕ. Furthermore, a h (u h , π h ϕ) = Ω G(u h )•G(π h ϕ) + j ′ h (u h , π h ϕ) = T 3 + T 4 .
Clearly,

T 3 → Ω ∇u•∇ϕ because of the strong convergence of G h (u h ) to ∇u in L 2 (Ω) d and the weak convergence of G h (π h ϕ) to ∇ϕ in L 2 (Ω) d . In addition, |T 4 | ≤ c|u h | J,F h ,-1 |ϕ -π h ϕ| J,F h ,-1 ≤ c ′ |ϕ -π h ϕ| J,F h ,-1 which converges to zero. As a result, a h (u h , π h ϕ) → Ω ∇u•∇ϕ.
The proof can now be completed as in the symmetric case.

Classical examples of the situation analyzed by Theorem 3.2 are the so-called Incomplete Interior Penalty method (IIPG) for which

(37) G h (v h ) = ∇ h v h ,
and the so-called Nonsymmetric Interior Penalty method (NIPG) for which

(38) G h (v h ) = ∇ h v h + R h ( v h ), together with G h (v h ) = ∇ h v h -R h ( v h ) in both cases.

The Stokes equations

Let f ∈ L r (Ω) d with r = 2d d+2 if d ≥ 3 and r > 1 if d = 2. Let ν > 0.
The components in the Cartesian basis (e 1 , . . . , e d ) of R d of a function, say v, with values in R d will be denoted by (v i ) 1≤i≤d . Implicit summation convention of repeated indices is adopted henceforth. Consider the Stokes equations (39)

         -ν∆u i + ∂ i p = f i , in Ω, i ∈ {1, . . . , d}, ∂ i u i = 0,
in Ω, u = 0, on ∂Ω, Ω p = 0. The weak formulation of this system consists of finding (u, p)

∈ H 1 0 (Ω) d × L 2 0 (Ω) s.t. for all (v, q) ∈ H 1 0 (Ω) d × L 2 0 (Ω), (40) ν
Ω ∂ j u i ∂ j v i - Ω p∂ i v i + Ω q∂ i u i = Ω f i v i .
The well-posedness of the above problem is a classical result (see e.g. [START_REF] Ern | Theory and Practice of Finite Elements[END_REF] and references therein).

To formulate a DG approximation, we consider for each component of the velocity the symmetric DG bilinear form a h defined by [START_REF] Eymard | Analysis tools for finite volume schemes[END_REF] and the stabilization bilinear form j h defined by [START_REF] Eymard | Convergence analysis of a colocated finite volume scheme for the incompressible navier-stokes equations on general 2D or 3D meshes[END_REF]. For the sake of simplicity, in particular with an eye towards ease of implementation, we will consider the case of equal-order polynomial interpolation for the velocity and for the pressure. Letting k ≥ 1, we thus set

(41) U h def = [V k h ] d , P h def = V k h /R, X h def = U h × P h .
For R d -valued functions such as velocities, the seminorm | • | J,F h ,-1 and the norm • DG are defined as the square root of the sum of the squares of the corresponding seminorm or norm for all the components. 4.1. Discrete divergence operators. Define on U h × P h the bilinear form

(42) b h (v h , q h ) def = Ω v h •∇ h q h - F ∈F i h F ν F •{ {v h } } q h .
Integration by parts readily yields the following equivalent expression

(43) b h (v h , q h ) = - Ω q h ∇ h •v h + F ∈F h F ν F • v h { {q h } }.
Here, ∇ h • denotes the broken divergence operator acting elementwise. Furthermore, define on P h × P h the pressure stabilization bilinear form (44)

s h (q h , r h ) def = F ∈F i h γh F F q h r h .
Here, γ ∈ R + is a penalty parameter. For simplicity, it will be taken equal to 1 in what follows. The basic stability result for the bilinear form b h is the following.

Lemma 4.1. There is β > 0, depending on Ω, k and P, such that

(45) ∀q h ∈ P h , β q h L 2 (Ω) ≤ sup 0 =v h ∈U h b h (v h , q h ) v h DG + |q h | J,F i h ,1 .
Proof. Let q h ∈ P h . Owing to a result by Nečas [START_REF] Nečas | Equations aux Dérivées Partielles[END_REF], there [START_REF] Cockburn | Local Discontinuous Galerkin methods for the Stokes system[END_REF], it is inferred that

is v ∈ H 1 0 (Ω) d s.t. ∇•v = q h and v H 1 (Ω) d ≤ c Ω q h L 2 (Ω) . Then, q h 2 L 2 (Ω) = Ω q h (∇•v) = - Ω ∇ h q h •v + F ∈F i h F q h { {v} }•ν F = - Ω ∇ h q h •π k h v + F ∈F i h F q h { {v} }•ν F = -b h (π k h v, q h ) + F ∈F i h F q h { {v -π k h v} }•ν F = T 1 + T 2 . Since π k h v DG ≤ c k,P v H 1 (Ω) d ≤ c Ω,k,P q h L 2 (Ω) because of
|T 1 | ≤ |b h (π k h v, q h )| π k h v DG π k h v DG ≤ c Ω,k,P sup 0 =v h ∈U h b h (v h , q h ) v h DG q h L 2 (Ω) . Similarly, |T 2 | ≤ c Ω,k,P |q h | J,F i h ,1 q h L 2 (Ω)
, whence the conclusion follows. Recall the discrete gradient operators G l h and G l h defined in §2.3. For all l ≥ 0, introduce now the discrete divergence operators D l h :

U h → V max(k-1,l) h defined s.t. ( 46
) ∀v h ∈ U h , D l h (v h ) def = G l h (v h,j
)•e j . For l ≥ k, the following integration by parts formula holds for all (v h , q h ) ∈ X h :

(47) Ω q h D l h (v h ) + Ω G l h (q h )•v h = 0.
Moreover, it is easily seen that for l ≥ k and for all (v h , q h ) ∈ X h ,

(48) b h (v h , q h ) = Ω v h •G l h (q h ) = - Ω q h D l h (v h ).
As before, superscripts will be dropped if l = k.

4.2.

Stability estimates and discrete well-posedness. For all ((u h , p h ), (v h , q h )) ∈ X h × X h , define the bilinear form

(49) l h ((u h , p h ), (v h , q h )) def = νa h (u h,i , u h,i ) + b h (v h , p h ) -b h (u h , q h ) + s h (p h , q h ).
The discrete Stokes equations consists of seeking (u h , p h ) ∈ X h s.t.

(50)

l h ((u h , p h ), (v h , q h )) = Ω f i v h,i , ∀(v h , q h ) ∈ X h .
Define the following norm

(51) (v h , q h ) 2 S def = v h 2 DG + |q h | 2 J,F i h ,1 + q h 2 L 2 (Ω) .
A direct consequence of ( 27) applied componentwise is the following result:

Lemma 4.2. Let α > 0 as in Lemma 3.2. Then, the following holds:

(52) ∀(v h , q h ) ∈ X h , να v h 2 DG + |q h | 2 J,F i h ,1 ≤ l h ((v h , q h ), (v h , q h ))
. Combining Lemmata 4.1 and 4.2 yields the following stability result.

Lemma 4.3.

There is c l > 0 depending on ν, k, P, Ω and η s.t.

(53) ∀(v h , q h ) ∈ X h , c l v h , q h ) S ≤ sup 0 =(w h ,r h )∈X h l h ((v h , q h ), (w h , r h )) (w h , r h ) S .
Proof. Let (v h , q h ) ∈ X h and set

S def = sup 0 =(w h ,r h )∈X h l h ((v h ,q h ),(w h ,r h )) (w h ,r h ) S . Owing to Lemma 4.2, να v h 2 DG + |q h | 2 J,F i h ,1 ≤ S (v h , q h ) S ,
and it only remains to control q h 2 L 2 (Ω) . Using Lemmata 4.1 and 3.1 yields

β q h L 2 (Ω) ≤ sup 0 =w h ∈U h b h (w h , q h ) w h DG + |q h | J,F i h ,1 ≤ sup 0 =w h ∈U h νa h (v h,i , w h,i ) w h DG + sup 0 =w h ∈U h l h ((v h , q h ), (w h , 0)) (w h , 0) S + |q h | J,F i h ,1 ≤ νc η,k,P v h DG + S + |q h | J,F i h ,1 .

The conclusion is straightforward.

A direct consequence of Lemma 4.3 is that for all h ∈ H, the discrete problem (50) admits a unique solution (u h , p h ) ∈ X h . 4.3. Convergence analysis. In this section, we are now interested in the convergence of the sequence {(u h , p h )} h∈H of solutions to the discrete Stokes equations (50) towards the unique solution (u, p) of the continuous Stokes equations (40). Theorem 4.1 (Convergence for Stokes equations). Let {(u h , p h )} h∈H be the sequence of approximate solutions generated by solving the discrete problems (50) on the admissible meshes {T h } h∈H . Then, as size(T h ) → 0,

u h → u, in L 2 (Ω) d , (54)
∇ h u h → ∇u, in L 2 (Ω) d,d , (55)
|u h | J,F h ,-1 → 0, ( 56
)
p h → p, in L 2 (Ω), ( 57
)
|p h | J,F i h ,1 → 0, ( 58
)
where (u, p) ∈ H 1 0 (Ω) × L 2 0 (Ω) is the unique solution to (40). Proof. (i) A priori estimates. Owing to the inf-sup condition (53), the assumption on f and the discrete Sobolev embedding, the sequence {(u h , p h )} h∈H is bounded in the • S -norm. Hence, up to a subsequence, there is (u, p)

∈ H 1 0 (Ω) × L 2 0 (Ω) s.t. u h → u strongly in L 2 (Ω) d , G h (u h,i ) ⇀ ∇u i weakly in L 2 (Ω) d
for all i ∈ {1, . . . , d}, and p h ⇀ p weakly in L 2 (Ω). (ii) Identification of the limit and convergence of the whole sequence. Let ϕ ∈ C ∞ c (Ω) d . Testing with (π h ϕ, 0) yields

νa h (u h,i , π h ϕ i ) + b h (π h ϕ, p h ) = Ω f i π h ϕ i .
Clearly, as size(T h ) → 0, the right-hand side tends to Ω f i ϕ i . Furthermore, proceeding as for the Poisson problem yields that the first term in the left-hand side converges to ν Ω ∂ j u i ∂ j ϕ i . Consider now the second term and observe that

b h (π h ϕ, p h ) = - Ω p h ∇ h •π h ϕ + F ∈F h F ν F • π h ϕ { {p h } } = T 1 + T 2 .
Owing to the weak convergence of {p h } h∈H to p in L 2 (Ω) and the strong convergence of {∇ h •π h ϕ} h∈H to ∇•ϕ in L 2 (Ω), T 1 tends to -Ω p(∇•ϕ). Moreover, using the trace inequality (3) to estimate { {p h } } L 2 (F ) yields

|T 2 | ≤ c k,P ϕ -π h ϕ DG p h L 2 (Ω) ≤ C ϕ -π h ϕ DG .
Hence, T 2 tends to zero. As a result,

ν Ω ∂ j u i ∂ j ϕ i - Ω p∂ j ϕ j = Ω f i ϕ i . Let now ψ ∈ C ∞ c (Ω)/R. Testing with (0, π h ψ) yields -b h (u h , π h ψ) + s h (p h , π h ψ) = 0. Using (48) yields -b h (u h , π h ψ) = Ω π h ψD h (u h ). Since {D h (u h )} h∈H weakly con- verges to ∇•u in L 2
(Ω) and {π h ψ} h∈H strongly converges to ψ in L 2 (Ω), the first term in the left-hand side tends to Ω ψ(∇•u). Furthermore, the second term tends to zero since

|s h (p h , π h ψ)| ≤ c k,P |p h | J,F i h ,1 |π h ψ| J,F i h ,1 ≤ C|π h ψ| J,F i h ,1 ,
and this upper bound tends to zero. Hence,

Ω ψ∂ j u j = 0. By density of C ∞ c (Ω) d × C ∞ c (Ω)/R in H 1 0 (Ω) d × L 2 0 (Ω)
, this shows that (u, p) solves the Stokes equations (40). Since the solution to this problem is unique, the whole sequence {(u h , p h )} h∈H converges. (iii) Strong convergence of the velocity gradient and convergence of velocity and pressure jumps. Observe that

Ω f i u h,i = l h ((u h , p h ), (u h , p h )) ≥ νa h (u h,i , u h,i ) + s h (p h , p h ) ≥ νa h (u h,i , u h,i ) ≥ d i=1 ν G h (u h,i ) 2 L 2 (Ω) d .
Thus, lim sup

d i=1 ν G h (u h,i ) 2 L 2 (Ω) d ≤ lim sup Ω f i u h,i = Ω f i u i = ν ∇u 2 L 2 (Ω) d,d . Since lim inf d i=1 G h (u h,i ) 2 L 2 (Ω) d ≥ ∇u 2 L 2 (Ω) d,d
owing to weak convergence, this classically implies the strong convergence in L 2 (Ω) d of G h (u h,i ) to ∇u i for all i ∈ {1, . . . , d}. The above inequalities also imply that a h (u h,i , u h,i ) → ∇u 2 L 2 (Ω) d,d , and proceeding as for the Poisson problem, it is deduced that

|u h | J,F h ,-1 → 0. Finally, since |p h | 2 J,F i h ,1 = b h (u h , p h ) = Ω f i u h,i -νa h (u h,i , u h,i ), it is inferred that |p h | J,F i h ,1 → 0. (iv)
Strong convergence of the pressure. Using again the result by Nečas [START_REF] Nečas | Equations aux Dérivées Partielles[END_REF], let

v(p h ) ∈ H 1 0 (Ω) d be s.t. ∇•v(p h ) = p h with v(p h ) H 1 (Ω) d ≤ c Ω p h L 2 (Ω) and set v h = π k h v(p h ).
Then, proceeding as in the proof of Lemma 4.1 yields

p h 2 L 2 (Ω) ≤ c Ω,k,P |p h | J,F i h ,1 p h L 2 (Ω) -b h (v h , p h ) ≤ c Ω,k,P |p h | J,F i h ,1 p h L 2 (Ω) + νa h (u h,i , v h,i ) - Ω f i v h,i = T 1 + T 2 -T 3 .
Since |p h | J,F i h ,1 tends to zero and p h L 2 (Ω) is bounded, T 1 converges to zero. Furthermore, since the sequence {v h } h∈H is bounded in the

• DG -norm because v h DG ≤ c k,P v(p h ) H 1 (Ω) d ≤ c Ω,k,P p h L 2 (Ω) , there is v ∈ H 1 0 (Ω) d such that, up to a subsequence, v h → v strongly in L 2 (Ω) d and G h (v h,i ) ⇀ ∇v i weakly in L 2 (Ω) d
for all i ∈ {1, . . . , d}. Owing to the uniqueness of the limit in the distribution sense, it is inferred that ∇•v = p. Consider now the terms T 2 and T 3 . It is clear that

T 3 → Ω f •v. Furthermore, T 2 = νa h (u h,i , v h,i ) = ν Ω G h (u h,i )•G h (v h,i ) + νj h (u h,i , v h,i ) = T 2,1 + T 2,2 .

Owing to the strong convergence of {G

h (u h,i )} h∈H in L 2 (Ω) d and to the weak convergence of {G h (v h,i )} h∈H in L 2 (Ω) d , it is inferred that T 2,1 → ν Ω ∂ j u i ∂ j v i . Moreover, |T 2,2 | ≤ c ν,k,P |u h | J,F h ,-1 |v h | J,F h ,-1 ≤ C|u h | J,F h ,-1 ,
which converges to zero. Collecting the above estimates leads to

lim sup p h 2 L 2 (Ω) ≤ ν Ω ∂ j u i ∂ j v i - Ω f i v i = Ω p∂ j v j = p 2 L 2 (Ω) ,
classically yielding the strong convergence of the pressure in L 2 (Ω).

Remark 4.1. If the exact solution (u, p) turns out to be more regular and belongs to the broken Sobolev space H k+1 (T h ) d × H k (T h ), one optimal a priori error estimates of the form (u-u h , p-p h ) S ≤ c u,p size(T h ) k can be established; see e.g. [START_REF] Cockburn | Local Discontinuous Galerkin methods for the Stokes system[END_REF][START_REF] Di | Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux[END_REF][START_REF] Ern | Discontinuous Galerkin methods for Friedrichs' systems. III. Multi-field theories with partial coercivity[END_REF].

The steady incompressible Navier-Stokes equations

In this section the space dimension is either 2 or 3. Let f ∈ L r (Ω) d with r =

         -ν∆u i + ∂ j (u i u j ) + ∂ i p = f i , in Ω, i ∈ {1, . . . , d}, ∂ i u i = 0,
in Ω, u = 0, on ∂Ω, Ω p = 0. The weak formulation of this system consists of finding (u, p)

∈ H 1 0 (Ω) d × L 2 0 (Ω) s.t. for all (v, q) ∈ H 1 0 (Ω) d × L 2 0 (Ω), (60) ν
Ω ∂ j u i ∂ j v i + Ω ∂ j (u i u j )v i - Ω p∂ i v i + Ω q∂ i u i = Ω f i v i .
The existence of a weak solution in the above sense, in two and three space dimensions, is a classical result; see, e.g., [START_REF] Temam | Theory and numerical analysis[END_REF][START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]. The uniqueness of the solution holds only under small data assumptions; see Remark 5.1 below.

5.1.

Design of the convective trilinear form. We choose the same discrete spaces for the velocity and for the pressure as for the Stokes equations. To allow for some generality in the treatment of the convective term, we introduce two parameters α 1 , α 2 ∈ {0, 1} and rewrite the momentum equation in the Navier-Stokes equations as (61)

-ν∆u i + ∂ j (u i u j ) -α 1 1 2 (∂ j u j )u i + α 2 1 2 ∂ i (u j u j ) + ∂ i p = f i , with the modified pressure (62) p def = p -α 2 1
2 (u j u j ). The choice (α 1 , α 2 ) = (1, 0) corresponds to Temam's device (see e.g. [START_REF] Temam | Theory and numerical analysis[END_REF]) to achieve stability. The choice (α 1 , α 2 ) = (0, 1) has been hinted to in [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF]; the modified pressure p differs from the Bernoulli pressure but the advantage is that the left-hand side of ( 61) is in divergence form, thereby lending itself to a conservative discretization.

Define on [H

1 0 (Ω) d ] 3 the trilinear form (63) t(w, u, v) def = Ω ∂ j (w i u j )v i -α 1 1 2 Ω (∂ j w j )u i v i + α 2 1 2 Ω ∂ i (w j u j )v i .
The discrete counterpart of the trilinear form t is a trilinear form t h defined on [U h ] 3 and for which the following design conditions are relevant.

(t1) For all v h ∈ U h , t h (v h , v h , v h ) = 0.
(t2) There is c t , depending on k and P, such that for all (w h , u

h , v h ) ∈ [U h ] 3 , t h (w h , u h , v h ) ≤ c t w h DG u h DG v h DG . (
v ∈∈ H 1 0 (Ω) d s.t. v h → v strongly in L 2 (Ω) d . Then, as size(T h ) → 0, t h (u h , u h , v h ) → t(u, u, v).

Discrete well-posedness and basic stability estimates. The discrete problem consists of seeking (u

h , p h ) ∈ X h s.t. (64) l h ((u h , p h ), (v h , q h )) + t h (u h , u h , v h ) = Ω f i v h,i , ∀(v h , q h ) ∈ X h ,
where the bilinear form l h associated with the Stokes equations is defined by (49).

In this section, the discrete trilinear form t h is assumed to satisfy (t1)-(t2) only.

Lemma 5.1 (A priori estimates). Let (u h , p h ) ∈ X h and assume that (u h , p h ) solves (64). Then, the following a priori estimates hold:

(να) 2 u h 2 DG + 2αν|p h | 2 J,F i h ,1 ≤ σ 2 r ′ f 2 L r (Ω) d , (65)
c l (u h , p h ) S ≤ σ r ′ f L r (Ω) d + c t (να) -2 (σ r ′ f L r (Ω) d ) 2 . ( 66
)
Proof. To prove (65), simply test (64) with (u h , p h ), observe that t h (u h , u h , u h ) = 0 owing to (t1) and use Lemma 4.2 for the linear part yielding

να u h 2 DG + |p h | 2 J,F i h ,1 ≤ Ω f i u h,i ≤ σ r ′ f L r (Ω) d u h DG ,
whence (65) is easily deduced. To prove (66), use the inf-sup condition in Lemma 4.3 and assumption (t2) to infer

c l (u h , p h ) S ≤ σ r ′ f L r (Ω) d + c t u h 2 DG ,
and conclude using (65).

To prove the existence of a discrete solution, we use a topological degree argument; see, e.g., [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF][START_REF] Eymard | Convergence analysis of a colocated finite volume scheme for the incompressible navier-stokes equations on general 2D or 3D meshes[END_REF] for the use of this argument in the convergence analysis of FV schemes and [START_REF] Deimling | Nonlinear functional analysis[END_REF] for a general presentation. Lemma 5.2. Let V be a finite dimensional functional space equipped with a norm • V , let µ > 0, and let Ψ : V × [0, 1] → V satisfying the following assumptions:

(i) Ψ is continuous; (ii) Ψ(•, 0) is an affine function and the equation Ψ(v, 0) = 0 has a solution v ∈ V such that v V < µ; (iii) For any (v, ρ) ∈ V × [0, 1], Ψ(v, ρ) = 0 implies v V = µ.
Then, there exists v ∈ V such that Ψ(v, 1) = 0 and v V < µ. Proposition 5.1. For all h ∈ H, the discrete problem (64) admits at least one solution (u h , p h ) ∈ X h . Proof. To apply Lemma 5.2, let V = X h and define the mapping Ψ :

X h × [0, 1] → X h such that for (u h , p h ) given in X h and ρ given in [0, 1], (ξ h , ζ h ) def = Ψ((u h , p h ), ρ) ∈ X h is defined such that for all (v h , q h ) ∈ X h , (ξ h , v h ) L 2 (Ω) d = l h ((u h , p h ), (v h , 0)) + ρt h (u h , u h , v h ) - Ω f i v h,i , (ζ h , q h ) L 2 (Ω) = l h ((u h , p h ), (0, q h )).
Observing that l h is continuous on X h × X h for the • S -norm, using (t2) and the equivalence of norms in finite dimension, it is inferred that Ψ is continuous. Furthermore, point (ii) in Lemma 5.2 results from the a priori estimate for the Stokes equations. In addition, because of (t1), if (u h , p h ) ∈ X h is such that Ψ((u h , p h ), ρ) = 0 for some ρ ∈ [0, 1], then (u h , p h ) is bounded independently of ρ. This concludes the proof.

Convergence analysis.

In this section, we are now interested in the convergence of a sequence {(u h , p h )} h∈H of solutions to the discrete problem (64) towards a solution (u, p) of the Navier-Stokes equations (60). The same convergence result can be established as for the Stokes equations. The only difference is that, because we do not make a smallness assumption on the data, there is no uniqueness result available at the continuous level, and thus only the convergence of subsequences (and not of the whole sequence) is obtained.

Theorem 5.1 (Convergence for Navier-Stokes equations). Let {(u h , p h )} h∈H be a sequence of approximate solutions generated by solving the discrete problems (64) on the admissible meshes {T h } h∈H . Assume (t1)-(t3). Then, as size(T h ) → 0, up to a subsequence,

u h → u, in L 2 (Ω) d , (67)
∇ h u h → ∇u, in L 2 (Ω) d,d , (68)
|u h | J,F h ,-1 → 0, (69)
p h ⇀ p, weakly in L 2 (Ω), (70)
|p h | J,F i h ,1 → 0, (71) where
(u, p + α 2 1 2 (u j u j )) ∈ H 1 0 (Ω) × L 2 0 (Ω) is a solution to (60). Moreover, if (t4) also holds, then p h → p in L 2 (Ω).
Proof. (i) Proceeding as for the Stokes equations, it is clear that there is (u, p) ∈

H 1 0 (Ω) × L 2 0 (Ω) s.t., up to a subsequence, u h → u strongly in L 2 (Ω) d , G h (u h,i ) ⇀ ∇u i weakly in L 2 (Ω) d for all i ∈ {1, . . . , d} and p h ⇀ p weakly in L 2 (Ω).
(ii) Identification of the limit. Using (t3) and proceeding as for the Stokes equations to treat the linear part, it is inferred that for all

ϕ ∈ C ∞ c (Ω) d , ν Ω ∂ j u i ∂ j ϕ i + t(u, u, ϕ) - Ω p∂ j ϕ j = Ω f i ϕ i .
and that for all

ψ ∈ C ∞ c (Ω)/R, Ω ψ∂ j u j = 0.
Hence, (u, p + α 2 1 2 (u j u j )) solves the incompressible Navier-Stokes equations. (iii) Strong convergence of the velocity and of the jumps. Proceeding as for the Stokes equations, (t1) yields the strong convergence of the piecewise velocity gradient in L 2 (Ω) d and the convergence to zero of the jump seminorms |u h | J,F h ,-1 and |p h | J,F i h ,1 . (iv) Strong convergence of the pressure. Proceeding as for the Stokes equations yields

p h 2 L 2 (Ω) ≤ c Ω,k,P |p h | J,F i h ,1 p h L 2 (Ω) + νa h (u h,i , v h,i ) + t h (u h , u h , v h ) - Ω f i v h,i = T 1 + T 2 + T 3 -T 4 .
The convergence of T 1 , T 2 and T 4 is treated as for the Stokes equations. Furthermore, the convergence of T 3 results from assumption (t4). As a result,

lim sup p h 2 L 2 (Ω) ≤ ν Ω ∂ j u i ∂ j v i + t(u, u, v) - Ω f i v i = Ω p(∂ i v i ) + α 2 1 2 Ω ∂ i (u j u j )v i = Ω p(∂ i v i ) = p 2 L 2 (Ω) ,
concluding the proof.

Remark 5.1. Under a smallness condition of the form

c Ω,k,P ν -2 f L r (Ω) d < 1,
uniqueness of the weak solution of (60) classically holds, so that the conclusions (67)-(71) of Theorem 5.1 apply to the whole sequence {(u h , p h )} h∈H . Moreover, the convergence of the fixed-point iterative scheme

l h ((u k+1 h , p k+1 h ), (v h , q h )) + t h (u k h , u k+1 h , v h ) = Ω f i v h,i , ∀(v h , q h ) ∈ X h ,
can be proven using standard arguments.

Examples. Define for (w

h , u h , v h ) ∈ [U h ] 3 , ( 72
)
t h (w h , u h , v h ) = Ω (w h •∇ h u h )•v h - F ∈F i h F { {w h } }•ν F u h •{ {v h } } + Ω 1 2 (∇ h •w h )(u h •v h ) - F ∈F h F w h •ν F 1 2 { {u h •v h } }.
This choice corresponds to (α 1 , α 2 ) = (1, 0). The resulting DG method is not conservative, but contains a source term proportional to the divergence of the discrete velocity (still converging to zero as the mesh is refined).

Proposition 5.2. Let t h be defined by (72). Then, assumptions (t1)-(t4) hold.

Proof. The verification of (t1) is straightforward. Assumption (t2) results from the Sobolev embedding with q = 4 and trace inequalities. To prove (t3) and (t4), observe first that for all v h ∈ U h ,

t h (u h , u h , v h ) = Ω u h •G 2k h (u h,i )v h,i + 1 4 F ∈F i h F u h,i ν F • u h v h,i + Ω D 2k h (u h ) 1 2 u h,i v h,i = T 1 + T 2 + T 3 .
To prove (t3), take v h = π h ϕ with ϕ ∈ C ∞ c (Ω) d . Owing to the discrete Sobolev embedding with q = 4, the sequences {u h } h∈H and {π h ϕ} h∈H are bounded in L 4 (Ω) d . Hence, Lebesgue's Dominated convergence Theorem implies that, up to a subsequence, u h π h ϕ i converges to uϕ i in L 2 (Ω) d . In addition, {G 2k h (u h,i )} h∈H weakly converges to ∇u i in L 2 (Ω) d . As a result, T 1 converges to Ω u j (∂ j u i )ϕ i . Similarly, T 3 converges to Ω 1 2 (∂ j u j )u i ϕ i . Furthermore, T 2 → 0 since |u h | J,F h ,-1 is bounded and π h ϕ i L ∞ (F ) converges to zero. Therefore, as size(T h ) → 0,

t h (u h , u h , π h ϕ) → Ω u j (∂ j u i )ϕ i + Ω 1 2 (∂ j u j )u i ϕ i = Ω [∂ j (u i u j ) -1 2 (∂ j u j )u i ]ϕ i .
Assumption (t4) is proven similarly for the terms T 1 and T 3 . To prove that T 2 converges to zero, observe that |u h | J,F h ,-1 converges to zero and that v h L ∞ (F ) ≤ c k,P h -1 F owing to a trace inequality. This concludes the proof.

Define now for (w

h , u h , v h ) ∈ [U h ] 3 , (73)
t h (w h , u h , v h ) = - Ω (w h,i u h •∇ h v h,i + F ∈F i h F ν F •{ {u h } }{ {w h,i } } v h,i + Ω 1 2 v h •∇ h (u h,i w h,i ) - F ∈F i h F ν F •{ {v h } } 1 2 u h,i w h,i .
This choice corresponds to (α 1 , α 2 ) = (0, 1). The salient feature of the resulting DG method is that it is locally conservative.

Proposition 5.3. Let t h be defined by (73). Then, assumptions (t1)-(t4) hold.

Proof. Assumptions (t1)-(t2) can be readily verified. To prove (t3) and (t4), observe that for all v h ∈ U h ,

t h (u h , u h , π h ϕ) = - Ω u h,i u h •G 2k h (v h,i ) - 1 4
F ∈F i h F ν F • u h u h,i v h,i - Ω 1 2 u h,i u h,i D 2k h (v h ),
where

D 2k h (v h ) def = G 2k h (v h,i )•e i .
Then, proceed as in the previous proof to infer that for all ϕ ∈ C ∞ c (Ω) d , as size(T h ) → 0,

t h (u h , u h , π h ϕ) → Ω [∂ j (u i u j ) + 1 2 ∂ i (u j u j )]ϕ i ,
along with a similar result for (t4). Remark 5.2. Upwinding can be introduced in the discrete trilinear forms t h defined by (72) or (73) by adding a term of the form

F ∈F i h θ F F |{ {w h } }•ν F | u h • v h ,
and replacing the design assumption (t1) by the requirement that t h be nonnegative, which is sufficient to derive all the necessary a priori estimates and the convergence result of Theorem 5.1. Here, the parameter θ F ∈ [0, 1] depends on the local Péclet number. 5.5. Numerical experiment. To verify the asymptotic convergence properties of the method defined by (72), we have considered the analytical solution proposed in [START_REF] Kovasznay | Laminar flow behind a two-dimensional grid[END_REF] on the square domain Ω def = (-0.5, 1.5) × (0, 2),

u 1 = 1 -e -πx2 cos(2πx 2 ), u 2 = - 1 2 e πx1 sin(2πx 2 ), p = - 1 2 e πx1 cos(2πx 2 ) -p,
where p def = 1 meas(Ω) Ω -1 2 e πx1 cos(2πx 2 ) ≃ -0.920735694 ensures zero-mean for the pressure, ν = 1 3π and f = 0. The example was run on a family of uniformly refined triangular meshes with mesh sizes ranging from 0.5 down to 0.03125, labeled with progressive numbers from 1 to 5 in Table 1. The nonlinear problem was solved by the exact Newton algorithm with tolerance set to 10 -6 ; the linear systems were solved using the direct solver available in PETSc. According to Table 1, the method converges with optimal order in the energy norm defined by (51). The method defined by (73) was also tested, and the corresponding asymptotic convergence rates were observed to be suboptimal by half an order. Further tests are out of the scope of the present paper and will receive extensive attention in a future work.

Discrete functional analysis in DG spaces

Let 1 ≤ p < +∞ and let k ≥ 1 be an integer. Equip the DG finite element space V k h defined by ( 4) with the norm

(74) v h p DG,p def = T ∈T h T |∇v h | p ℓ p + F ∈F h 1 h p-1 F F | v h | p , where |•| ℓ p denotes the ℓ p -norm in R d so that |∇v h | p ℓ p = d i=1 |∂ i v h | p . Recall that Ω is a open bounded connected subset of R d (d > 1)
whose boundary is a finite union of parts of hyperplanes. In this section, the mesh family {T h } h∈H used to build the DG spaces is assumed to satisfy only assumptions (i)-(iv) in Definition 2.1.

The material contained in this section, which is closely inspired from that derived in [START_REF] Eymard | Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes[END_REF] for discrete spaces of piecewise constant functions, deals with the extension to DG spaces of two key results of functional analysis, namely Sobolev embeddings and compactness criteria in L p (Ω). These results are presented here in a non-Hilbertian setting which is more general than that needed to analyze the Navier-Stokes equations. We have made this choice because the results below are of independent interest to analyze other nonlinear problems. We also observe that we deal here with functional analysis in DG spaces and not in broken Sobolev spaces. Lemma 6.1. For all 1 ≤ s < t < +∞, the following holds for all

v h ∈ V k h , (
′ = π π-1 yields v h s DG,s = T ∈T h T |∇v h | s ℓ s + F ∈F h 1 h s-1 F F | v h | s ≤ T ∈T h T d 1 π ′ |∇v h | s ℓ t + F ∈F h F h 1 π ′ F h 1 π (1-t) F | v h | s ≤ T ∈T h d T 1 π ′ 1 π ′ T ∈T h T |∇v h | t ℓ t 1 π + F ∈F h h F F 1 π ′ 1 π ′ F ∈F h 1 h t-1 F F | v h | t 1 π ≤ ((d + ̺ 1 )|Ω|) 1 π ′ v h s DG,t , using (1), whence the conclusion follows. Lemma 6.2. For v ∈ L 1 (R d ), define v BV = d i=1 sup{ R d u∂ i ϕ; ϕ ∈ C ∞ c (R d ), ϕ L ∞ (R d ) ≤ 1},
and set BV = {v ∈ L 1 (R d ); v BV < +∞}. Then, extending discrete functions in V k h by zero outside Ω, there holds V k h ⊂ BV and for all 1 ≤ p < +∞, (76)

∀v h ∈ V k h , v h BV ≤ c d,̺1,|Ω|,p v h DG,p .
Proof. Clearly, owing to Lemma 6.1, it suffices to prove (76) for p = 1. Integrating by parts, it is clear that for all v h ∈ V k h and for all ϕ

∈ C ∞ c (R d ) with ϕ L ∞ (R d ) ≤ 1, R d v h ∂ i ϕ = - R d (e i •∇ h v h )ϕ + F ∈F h F e i •ν F v h ϕ ≤ v h DG,1 .
Hence, v h BV ≤ d v h DG,1 , completing the proof.

Remark 6.1. In this section we could have allowed the case k = 0, although the derived results are not as interesting as for k ≥ 1 because • DG,p is not the natural norm with which to equip the space V 0 h when working with FV approximations to nonlinear second-order PDE's. Indeed, on V 0 h , the first term on the right-hand side of (74) (the broken gradient) drops out, and this entails that a length scale different from h F must be used for the jump term, thereby also requiring an additional (mild) assumption on the mesh family; see [START_REF] Eymard | Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes[END_REF] for the analysis in this case. Remark 6.2. The observation that the • DG,2 -norm controls the BV norm can also be found in [START_REF] Lew | Optimal BV estimates for a discontinuous Galerkin method for linear elasticity[END_REF] in the framework of linear elasticity. 6.1. Discrete Sobolev embeddings. Theorem 6.1 (Discrete Sobolev embeddings). For all q such that (

i) 1 ≤ q ≤ p * def = pd d-p if 1 ≤ p < d; (ii) 1 ≤ q < +∞ if d ≤ p < +∞; there is σ q,p such that (77) ∀v h ∈ V k h , v h L q (Ω) ≤ σ p,q v h DG,p .
The constant σ q,p additionally depends on k, |Ω|, and P. In particular, for the choice q = p which is always possible,

(78) ∀v h ∈ V k h , v h L p (Ω) ≤ σ p,p v h DG,p .
Proof. We follow L. Nirenberg's proof of Sobolev embeddings. (i) The case p = 1. Set 1 * def = d d-1 . Then, owing to a classical result (see, e.g. [START_REF] Eymard | Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes[END_REF] for a proof), for all v ∈ BV,

v L 1 * (R d ) ≤ 1 2d v BV .
Extending discrete functions in V k h by zero outside Ω, Lemma 6.2 yields

(79) v h L 1 * (R d ) ≤ 1 2 v h DG,1 ,
i.e., (77) for p = 1 and q = 1 * with σ 1,1 * = 1 2 , and hence for all 1 ≤ q ≤ 1 * since Ω is bounded. (ii) The case 1 < p < d. Set α = p(d-1) d-p and observe that α > 1. Considering the function |v h | α (extended by zero outside Ω) and using (79) yields (80) 2

Ω |v h | p * d-1 d ≤ T ∈T h T |∇|v h | α | ℓ 1 + F ∈F h F | |v h | α | ≡ T 1 + T 2 .
Observe that a.e. in each

T ∈ T h , |∂ i |v h | α | = α|v h | α-1 |∂ i v h | for all i ∈ {1, . . . , d} so that |∇|v h | α | ℓ 1 = α|v h | α-1 |∇v h | ℓ 1
. Using Hölder's inequality with p and q = p p-1 , the first term in (80) is bounded as

|T 1 | ≤ α T ∈T h T |v h | q(α-1) 1 q T ∈T h T |∇v h | p ℓ 1 1 p ≤ αd p-1 p Ω |v h | p * 1 q T ∈T h T |∇v h | p ℓ p 1 p
.

Furthermore, observing that | |v h | α | ≤ 2α{ {|v h | α-1 } }| v h | and using again Hölder's inequality, it is inferred that the second term in (80) is bounded as v h DG,p .

|T 2 | ≤ α T ∈T h F ⊂∂T F h 1 q F |v h|T | α-1 h -1 q F | v h | ≤ α T ∈T h F ⊂∂T F h F |v h|T | p * 1 q T ∈T h F ⊂∂T 1 h p-1 F F | v h | p 1 p ≤ α2 1 p τ 1 q p * ,k Ω |v h | p * 1 q F ∈F h 1 h p-1 F F | v h | p
Observing that d-1 d -1 q = 1 p * yields (77). (iii) The case d ≤ p < +∞. Fix any q 1 such that p < q 1 < +∞ and set p 1 = dq1 d+q1 so that p 1 < d and p * 1 = q 1 . Then, owing to point (ii) in this proof, it is inferred that for all v h ∈ V k h , v h L q 1 (Ω) ≤ σ p1,q1 v h DG,p1 , and the conclusion follows from Lemma 6.1 since p 1 ≤ p. 6.2. Compactness. In this section we are interested in sequences {v h } h∈H in V k h which are bounded in the • DG -norm. Theorem 6.2 (Compactness). Let {v h } h∈H be a sequence in V k h and assume that this sequence is bounded in the • DG,p -norm. Then, the family {v h } h∈H is relatively compact in L p (Ω) (and also in L p (R d ) taking v h = 0 outside Ω).

Proof. Extending the functions v h by zero outside Ω and observing that (see, e.g. [START_REF] Eymard | Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes[END_REF]) for all ξ ∈ R d ,

v h (• + ξ) -v h L 1 (R d ) ≤ |ξ| ℓ 1 v h BV ≤ C|ξ| ℓ 1 ,
because of the boundedness of the sequence {v h } h∈H in the • DG,p -norm (and hence in the BV-norm owing to Lemma 6.2), Kolmogorov's Compactness Criterion yields that the family {v h } h∈H is relatively compact in L 1 (R d ). Owing to the Sobolev embedding (78), this sequence is also bounded in L p (R d ); hence, it is also relatively compact in L p (R d ). Finally, the relative compactness also holds in L p (Ω) since the functions v h have been extended by zero outside Ω. Theorem 6.3 (Regularity of the limit). Let 1 < p < +∞. Let {v h } h∈H be a sequence in V k h and assume that this sequence is bounded in the • DG,p -norm. Assume that size(T h ) → 0. Then, there exists v ∈ W 1,p 0 (Ω) such that, up to a subsequence, v h → v in L p (Ω).

Proof. Owing to Theorem 6.2, there is v ∈ L p (Ω) such that, up to a subsequence, {v h } h∈H converges to v in L p (Ω). It remains to prove that v ∈ W 1,p 0 (Ω). To this purpose, we again extend the functions v h by zero outside Ω and we construct a discrete gradient converging, at least in the distribution sense over R d , to ∇v.

(1) Consider the lifting operators r 0 F and R 0 h defined in §2.3 and recall that the support of r 0 F consists of the one or two mesh elements of which F is a face. Hence,

R 0 h ( v h ) p L p (Ω) d = T ∈T h T F ⊂∂T r 0 F ( v h ) p ℓ p ≤ T ∈T h T N p-1 ∂ F ⊂∂T |r 0 F ( v h )| p ℓ p = N p-1 ∂ F ∈F h r 0 F ( v h ) p L p (Ω) d .
Furthermore, setting for all i ∈ {1, . . . , d}, y h,i = |r 0 F,i ( v h )| p-2 r 0 F,i ( v h ), observing that y h ∈ [V 0 h ] d and using Hölder's inequality with p and q = p p-1 yields

r 0 F ( v h ) p L p (Ω) d = Ω y h •r 0 F ( v h ) = F { {y h } }•ν F v h ≤ 2 -1 q   T ;F ⊂∂T h F F |y h|T •ν F | q   1 q 1 h p-1 F F | v h | p 1 p ≤ 2 -1 q   T ;F ⊂∂T h F d q p F |r 0 F ( v h )| p ℓ p   1 q 1 h p-1 F F | v h | p 1 p ≤ c d,p,k,P r 0 F ( v h ) p q L p (Ω) d 1 h p-1 F F | v h | p 1 p .
Collecting the above bounds yields

R 0 h ( v h ) L p (Ω) d ≤ c d,p,k,P F ∈F h 1 h p-1 F F | v h | p 1 p
.

Then, upon defining the approximate gradient

G 0 h (v h ) = ∇ h v h -R 0 h ( v h ) ∈ [V k h ]
R d G 0 h (v h )•ϕ = - R d v h (∇•ϕ) - R d R 0 h ( v h )•(ϕ -π 0 h ϕ) + F ∈F h F { {ϕ -π 0 h ϕ} }•ν F v h = T 1 + T 2 + T 3 .
Letting size(T h ) → 0, we observe that T 1 → -R d v(∇•ϕ) and that T 2 → 0 since ϕ-π 0 h ϕ L ∞ (R d ) d → 0 and R 0 h ( v h ) L p (R d ) d is bounded. Furthermore, proceeding as usual with q = p p-1 yields

T 3 ≤ c P ϕ -π 0 h ϕ L ∞ (R d ) d |Ω| 1 q F ∈F h 1 h p-1 F F | v h | p 1 p ≤ C ϕ -π 0 h ϕ L ∞ (R d ) d
whence it is inferred that T 3 → 0. As a result,

R d w•ϕ = lim size(T h )→0 R d G 0 h (v h )•ϕ = - R d v(∇•ϕ).
Hence, w = ∇v so that v ∈ W 1,p (Ω), and since v is zero outside Ω, v ∈ W 1,p 0 (Ω). Remark 6.3. For p = 2, lifting operators using a higher polynomial degree l ≥ 1 can also be considered as in the proof of Theorem 2.2. The difficulty for p = 2 is that the vector y h in the above proof is not necessarily polynomial-valued.

Proposition 2 . 1 (

 21 Stability of discrete gradients). Let k ≥ 1 and let l ≥ 0. Then,

6 5

 6 if d = 3 and r > 1 if d = 2. Let ν > 0. Consider the steady incompressible Navier-Stokes equations in conservative form (59)

Figure 2 .

 2 Figure 2. Plot of Kovasznay's solution for k = 1 and mesh 5.

,F 2 Ω

 2 where for s ∈ R + , τ s,k is the constant in the trace inequality∀ζ ∈ P k (T ), |ζ| s ≤ τ s,k T |ζ| s ,valid uniformly for all h ∈ H and for all T ∈ T h . This leads to|v h | p *

  d and extending it by zero outside Ω, it is inferred thatG 0 h (v h ) L p (R d ) d ≤ c d,p,k,P v h DG,p . Hence, the sequence {G 0 h (v h )} h∈H is bounded in L p (R d ) d , and thus since p > 1, up to a subsequence, G 0 h (v h ) ⇀ w weakly in L p (R d ) d . (ii) Let ϕ ∈ C ∞ c (R d ) dand observe that

2 .

 2 The discrete setting 2.1. Meshes. Let Ω be an open bounded connected subset of R d (d > 1) whose boundary is a finite union of parts of hyperplanes. Definition 2.1 (Admissible meshes). Let H be a countable set. The family {T h } h∈H is said to be an admissible mesh family if the following assumptions are satisfied: (i) for all h ∈ H, T h is a finite family of non-empty connex (possibly non-convex) open disjoint sets T forming a partition of Ω and whose boundaries are a finite union of parts of hyperplanes; (ii) there is a parameter N ∂ , independent of h, such that each T ∈ T h has at most N

∂ faces. A set F ⊂ ∂T is said to be a face of T is F is part of a hyperplane, and if either F is located on the boundary of Ω or there is

  t3) Let {u h } h∈H be a sequence in U h , bounded in the • DG -norm and such that there is u ∈ H1 0 (Ω) d s.t. u h → u strongly in L 2 (Ω) d and, for all i ∈ {1, . . . , d}, G h (u h,i ) ⇀ ∇u i weakly in L 2 (Ω) d . Then, for all ϕ ∈ C ∞ c (Ω) d , as size(T h ) → 0, t h (u h , u h , π h ϕ) → t(u, u, ϕ).Assume furthermore that, for all i ∈ {1, . . . , d}, G h (u h,i ) → ∇u i strongly in L 2 (Ω) d and that |u h | J,F h ,-1 → 0. Let {v h } h∈H be another sequence in U h , bounded in the • DG -norm and such that there is

	(t4)

Table 1 .

 1 Convergence results for the trilinear form defined by (72). We have set e h = (e h,u , e h,p ) def = (u -u h , p -p h ).

	mesh	h	e h,u L 2 (Ω) d order e h,p L 2 (Ω) order	e h S	order
	1	5.00e -1 8.87e -01	-	1.62e + 00	-	1.19e + 01	-
	2	2.50e -1 2.39e -01	1.89 6.11e -01 1.41 7.26e + 00 0.71
	3	1.25e -1 5.94e -02	2.01 2.01e -01 1.60 3.68e + 00 0.98
	4	6.25e -2 1.59e -02	1.90 7.40e -02 1.44 1.85e + 00 0.99
	5	3.12e -2 4.17e -03	1.93 3.14e -02 1.23 9.25e -01 1.00

Institut Franc ¸ais du Pétrole, 1 & 4, avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex -France

Université Paris-Est, CERMICS, Ecole des Ponts, Champs sur Marne, 77455 Marne la Vallée Cedex 2, FranceE-mail address: daniele-antonio.di-pietro@ifp.fr E-mail address: ern@cermics.enpc.fr

Acknowledgment. The authors are thankful to R. Eymard (University Paris-Est) for fruitful discussions.


.02

Context o

Increasing demand in poultry meat worldwide (+2%/year) o Rapid evolution of the market share 1 9 6 2 1 9 6 5 1 9 6 8 1 9 7 1 1 9 7 4 1 9 7 7 1 9 8 0 1 9 8 3 1 9 8 6 1 9 8 9 1 9 9 2 1 9 9 5 1 9 9 8 2 0 0 1 2 0 0 4 2 0 0 7 2 0 1 0 2 0 1 3

(

(oo(oooo

  protein content(Baéza et al., 2012; Janisch et al., 2013) -Modified post-mortem metabolism(Baéza et al., 2012) protein content(Baéza et al., 2012; Janisch et al.Alnahhas et al., 2014,2015) M. DUCLOS & C. BERRI / From muscle to meat : physiological and metabolic basis .015 Metabolomic profile (muscle & serum) ♂ pHu-/pHu+ (n = 40) [6.3 vs 5.5 -glycogen 195 vs 145] Polar extract of P. major muscle (1H et 31P NMR): 25 discriminant metabolites Polar extract of serum: (1H NMR): 20 discriminant metabolites Serum (Beauclercq et al., J. Proteome Res. 2016, 15, 1168-1178) M. DUCLOS & C. BERRI / From muscle to meat : physiological and metabolic basis .016 pHu -: Overrepresentation of metabolites related to carbohydrate turn-over pHu +: Overrepresentation of metabolites related to oxidation and protein catabolism Enrichment analysis on discriminant metabolites Metabolomic profile (muscle& serum) M. DUCLOS & C. BERRI / From muscle to meat : physiological and metabolic basis .017 (Beauclercq et al., 2016) Metabolomic profile (muscle & serum) M. DUCLOS & C. BERRI / From muscle to meat : physiological and metabolic basis Low blood supply → low cell oxygenation, low nutrient supply o Intensive use of in situ glycogen o Elevated protein and muscle degradation o Increased oxidative stress and production of reactive oxidative species (ROS) → altered redox homeostasis o Muscle dysfunction? Fiber alterations? Link with emergent defects? Global gene expression analysis by RNA-sequencing: o Abnormal accumulation of intracellular calcium o Hypoxia o Increased oxidative stress and accumulation of ROS o Fiber-type switching (fast to slow) o Increased cellular repair and regeneration in skeletal muscle o Downregulation of genes related to coagulation system Genomics and metabolomics are very helpful to unravel meat quality determinism and identify pertinent biomarkers for diagnosis purpose o It highlights some common regulatory pathways underlying meat defects related to muscle growth and glycogen depletion… o Poor muscle fiber vascularity → poor oxygenation of the muscle, anaerobic glycolysis, oxidative stress o Production of ROS (↓ redox homeostasis) Developing non invasive methods to identify breeders prone to develop meat defects: Genetic markers useful for selection (genomics) o Biomarkers such as blood metabolite, enzyme activity, or gene expression o Ensuring an optimal and harmonious development of broilers with positive impact on animal robustness and meat quality: o Livestock management, especially during early life o Preserve glycogen store in muscle, o Delay the differentiation processes to produce more fibers of smaller size o Increase the pool of available satellite cells to insure proper cell growth and renewal (limit fibrosis and lipidosis) o Considering the use of more robust strains with more harmonious body development:Gains expected in term of mortality, meat waste at the slaughter plant,