Prof Alonso Ana

Rodríguez

Keywords: Formes de Whitney d'ordre supérieur, éléments finis d'ordre supérieur, poids, petits simplexes High order Whitney forms, high order finite elements, weights, small simplices I

Titre : Des poids comme degrés de liberté pour les éléments finis de Whitney d'ordre élevé

Résumé :

Les formes de Whitney d'ordre élevé, également connues sous le nom de formes différentielles polynomiales tronquées, sont une célèbre famille de formes différentielles. Ils trouvent leurs racines dans le livre de Hassler Whitney, Théorie de l'intégration géométrique, publié en 1957, où une contrepartie de faible degré a été utilisée pour prouver le célèbre théorème de Rham. Ce n'est que dans les années 80 qu'ils ont été reconnus comme un puissant outil d'analyse numérique, lorsqu'il a été prouvé qu'ils paramétraient la première famille d'éléments finis de Nédélec. Dans l'esprit de Whitney, on peut choisir des poids, à savoir des intégrales de k-formes sur des k-simplexes, comme degrés de liberté pour ces espaces. Pour ce faire, la notion de petits simplexes sera introduite. Un petit simplexe est un morceau d'un partitionnement (virtuel) d'un simplexe. Nous montrons que les poids associés aux petits simplexes appropriés assurent l'unisolvabilité de la première famille de Nédélec et nous proposons une stratégie pour façonner la géométrie des petits simplexes. Cela permet de généraliser les concepts classiques propres aux éléments finis lagrangiens et à l'interpolation lagrangienne à des cadres de dimension supérieure. Nous développons d'abord la théorie correspondante, puis proposons des exemples numériques, mettant en relation les résultats avec la géométrie du problème.

Contents

Introduction

Nédélec first family of finite elements [START_REF] Nédélec | Mixed finite elements in R 3[END_REF] was introduced in the '80s by French mathematician Jean-Claude Nédélec. His approach was purely analytical and this made it possible to systematically treat the features of these elements. This family gained a lot of popularity in several applications, in particular in electromagnetism. A few years later another French mathematician, Alain Bossavit, proposed an alternative way to deal with some issues arising from geometrical aspects of Nédélec's work. He proposed his ideas in several papers, such as [START_REF]Differential forms and the computation of fields and forces in electromagnetism[END_REF], and gathered them in the book Computational electromagnetism [START_REF]Computational electromagnetism[END_REF]. One of the greatest insights of Bossavit was reinterpreting Whitney's [START_REF] Whitney | Geometric Integration Theory[END_REF] and Weil's [START_REF] Weil | Sur les théorèmes de de Rham[END_REF] results on cohomology theory. We owe to his works the introduction of Whitney forms into computational electromagnetism and this opened the way to the introduction of differential forms in several others branches of numerical analysis and finite elements in particular [START_REF] Hiptmair | Canonical construction of finite elements[END_REF]. A unified treatise which also relates Nédélec first and second family [START_REF]A new family of mixed finite elements in R 3[END_REF] appeared at the beginning of this century [START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF]. This also made it possible to free the discussion from dimensional constraints, and Nédélec first family was then extended to an arbitrary dimension n 1 [ 3 5 ] .

Whitney forms offer a beautiful and solid bridge between geometry and numerical analysis. In fact, they relate each subsimplex of a simplex T to a differential form and, more specifically, to a degree one polynomial differential form ! .I t turns out that the vector space spanned by the collection of Whitney forms associated with the k-subsimplices ∆ k (T )ofthesimplexT is a subspace of degree one polynomial differential forms. This space is generally denoted by P 1 Λ k (T ). In the majority of the situations, such a subspace is proper, i.e. there is a strict inclusion P 1 Λ k (T ) ⇢P 1 Λ k (T ). A stunning feature of this space is that any of its elements is completely characterised by integration over the collection of k-subsimplices. Roughly speaking, if one knows the integral of a Whitney k-form on all k-faces of the generating simplex, then one indeed knows that form. Of course, one would be interested to provide a generalisation of these objects, avoiding the restriction on the polynomial degree and being able at the same time to mimic such a feature. One thus aims to produce a definition for high order Whitney forms P r Λ k (T ), i.e. spaces that boil down to usual Whitney forms if the polynomial degree is one, vii that makes them compatible with this structure offered by integration. Their most common definition, which justifies the widely diffused name of trimmed polynomial differential forms for these spaces, involves an algebraic tool, the Koszul differential  [START_REF] Lang | ofGraduateT extsinMathematics[END_REF], which does not offer an immediate generalisation of the geometrical features. To recover them, in this thesis I shall follow [START_REF] Rapetti | Whitney forms of higher degree[END_REF] and see the spaces of trimmed polynomials differential forms as the product of polynomial spaces P r (T ) and Whitney forms in P 1 Λ k (T ). This makes it possible to characterise a monomial system of generators for such spaces, then a combinatorial result is invoked to identify an appropriate basis. On this basis one can reproduce a geometrical duality with the aim of a (virtual) finer partition of the simplex. It is important to stress that such a subdivision is virtual,i nt h es e n s et h a to n en e v e rr e fi n e so r places a finer mesh on T .T h i si sac r u c i a la s p e c t .I nt h i st h e s i sIs h o wb o t hc o nstructions, exploiting the power of the algebraic approach and relating the second one with the geometry of the element. These features will be extensively used throughout the whole work.

In the context of finite elements, a crucial role is played by the definition of degrees of freedom for shape functions. Their role is twofold: on the one hand, they make it possible to interpolate locally functions and fields, and on the other they provide gluing conditions for adjacent domains that share a part of the boundary. They also provide the natural definition of a projector onto the finite dimensional space by simply asking that continuous and discrete quantities have the same degrees of freedom. Either in its classical presentation or in the language of differential forms [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF], moments are widely spread as such degrees of freedom. When ! is a k-form, its moments are integrals over `-subsimplices of T ,w i t h`= k, . . . , n. This choice is perfectly meaningful, however it may leave unsatisfied under few aspects. For instance, if one thinks of problems arising from physics and electromagnetism, they would like to be able to use measurements, such as circulations of electric fields of fluxes of magnetic fields, as degrees of freedom. In order to link such data with moments there is some work to do.

The main topic of this work concerns thus the construction of alternative degrees of freedom for Nédélec first family of finite elements and it is cast in the language of differential forms, in arbitrary dimension n.Ih e n c ef o c u so nl o c a l spaces that constitute simplicial finite elements [START_REF] Ern | Finite Elements I: Interpolation and approximation[END_REF]. These claimed degrees of freedom are called weights,a n dh a v eb e e nf o r m a l i s e di n[ 6 0 ] . T h e yc o n s i s t so f integration of k-forms on specific k-simplices placed in appropriate positions in the domain. Such regions are called small simplices.T h em a i na d v a n t a g eo ft h i s approach is that any choice of small simplices gives a different interpolator, hence stability may be improved just by considering different weights, which consists in changing the small simplices on which integration is performed. For this reason, after having reviewed the existing literature, I will abandon usual small simplices viii and move toward nonuniform simplices.

The first known set of small simplices was built in [START_REF] Rapetti | Whitney forms of higher degree[END_REF] and it is referred to such af a m i l ya st h a to funiform small simplices. Such a construction finds its roots in [START_REF]High order edge elements on simplicial meshes[END_REF], where the relationship between generating functions for P r Λ k (T )a n ds o m e geometrical entities, eventually recognised as small simplices, were investigated. It is worth noting that, at this stage, spaces of trimmed polynomial differential forms were still treated in terms of their vector proxies. Since the number and dimension of small simplices depends on the polynomial degree r and the order k of the form, it is here denoted by X k r (T ). Its elements are the k-faces of small n-simplices that are 1 r -homothetic to T and whose vertices are anchored to specific points that are prescribled by the principal lattice.I n[ 6 0 ]a u t h o r sn o t i c e dad i ff e r e n c eb e t w e e nt h e cardinality of small simplices and the dimension of spaces of trimmed polynomial differential forms. They constructed a matrix, which I will formalise in this work under the name of generalised Vandermonde matrix,w h i c ha l l o w e dt or e a ds u c h ar e d u n d a n c ya n dh e n c et oi d e n t i f yt h ee x c e e d i n gs m a l ls i m p l i c e s . Af o r m a la n d general proof for unisolvence of such a family of small simplices was given in [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF]. Five years later, in [1], it was offered a detailed explanation and an algorithm to get rid of the redundant small simplices. Such a procedure required the introduction of some auxiliary sets, which I here retrace and denote by Σ k r (T ), that first break the desired geometrical structure of simplicial complex,t h e nt r a n s f o r mt h ep r o b l e m into a matter of linear algebra, and finally give in turn a subset X k r,min (T )o ft h e small simplices that is still unisolvent and that bears the desired cardinality. It is worth noting that proofs require some rigidity on the set of supports. Such hypotheses are related to the geometry of the problem, and in particular constrain to cases where some parallelism is present.

The capability of constructing new sets of supports, hence of degrees of freedom, opens the way to considering a very large class of weights, since any different set of supports gives a different family of weights. This extends the idea of varying points in Lagrangian interpolation. The point of view is then moved to obtaining new sets of small simplices as transformations of uniform ones. Of course, a careful choice is needed when choosing the correct mapping that defines these new sets, since unisolvence is not a priori granted. In fact, very few can be said when such am a pi sn o tag l o b a la ffi n i t y . S i n c ec o n fi n i n go u r s e l v e st os u c hac a s ew o u l db e restrictive, one looks for something more general. A key role is then played by simplicial isomorphisms,i . e . m a p p i n g st h a tp r e s e r v e ,i na na p p r o p r i a t es e n s e ,t h e simplicial geometry of the domain. For this reason, an equivalent condition to check unisolvence and minimality is provided. It comes again from linear algebra and consists in checking the invertibility of the generalised Vandermonde matrix associated with the desired set of small simplices. Several sets have then been constructed over nodes that are suitable for nodal interpolation, such as those ix presented in [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF] and [START_REF] Blyth | A Lobatto interpolation grid over the triangle[END_REF], and a computational proof of unisolvence is provided.

The subsequent natural step is to investigate the quality of a given family of small simplices, hence of weights. The suitable tool to do this is the generalised Lebesgue constant Λ r ,in troduc e din [START_REF] Alonso Rodríguez | On a generalization of the Lebesgue's constant[END_REF]. Suc haquan tit yre quire sthec onstruc tion of the cardinal basis, which is a basis for the underlying polynomial space that generalises the Lagrangian basis. Except in very few cases, an explicit computation requires the inversion of the generalised Vandermonde matrix. In accordance with the classical one, this new concept of Lebesgue constant still offers a bound on the stability of the interpolation and the local interpolation error. For this reason it is atrustfulindicatorofthequalityoftheweightsunderconsideration. V eryrecently , some tests have been performed in the three-dimensional context for edges [3] and faces [2]. In view of such analyses, a striking difference between the classical and the generalised Lebesgue constant emerged. In particular, both theoretical and computational results shows that the first is independent on the domain, i.e. it reads only the relative placement of nodes inside the domain, whereas the second is not, except in some very special cases. This is studied in details here and bounds are provided. This thesis is organised as follows. Chapter 1 is devoted to setting the notation and recalling basic definitions about discrete exterior calculus [START_REF] Hirani | Discrete exterior calculus[END_REF] and finite element exterior calculus [START_REF] Arnold | Finite element exterior calculus[END_REF], involving simplices, polynomial differential forms and Whitney forms. In this first chapter I offer a construction of both Whitney forms (the above claimed subspace of degree one polynomial differential forms) and the corresponding high order generalisations. Most of the results are proved, especially those that are frequently invoked. They are adapted to the language that I shall adopt throughout the whole work. The first chapter contains no new results, but ar e v i e wo ft h ee x i s t i n gm a t e r i a l ,g a t h e r e di nac o n v e n i e n tf a s h i o na n du n i f o r m e d in the notation of the whole work.

Chapter 2 is substantially divided into two parts. The first one is dedicated to retracing already known results about small simplices and their weights, such as those contained in [START_REF] Rapetti | Whitney forms of higher degree[END_REF], [START_REF]High order edge elements on simplicial meshes[END_REF], [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF] and [1]. I start from the definition of small simplices and characterise elements of such a set in terms of pairs. These pairs are given by a k-face of T and a multi-index. They give the position of a small simplex inside the "big" simplex with respect to its faces. In particular, the small simplex will be parallel to the k-face whereas the multi-index will represent the translation towards vertices of the simplex. Not only is this of independent interest, but it also allows for a neat relationship with the space of polynomial differential forms. This characterisation turns out to be very useful also for proving results about small simplices. Then known results about unisolvence are retraced. A discussion on some extremal case, i.e. that of points and of top-dimensional simplices, is enlightened. In the second part of Chapter 2, such results are sharpened and adapted x to the case of fish scale simplices, an auxiliary set which makes it possible to extract a minimal set from the preceding one. To be more definite, I first introduce this new construction, prove its features, such as unisolvence and minimality, and then exploit it to identify a subset of that of uniform small simplices which is still unisolvent and minimal. The last part of the chapter presents some application and special cases. In particular, it is shown that when either the dimension or the codimension of the set of small simplices is one, some simplification occurs. This is interesting, but may also be misleading: in fact, in the three dimensional space one could treat all possible cases by adopting these particular considerations, which would hide the underlying general technique here adopted.

Chapter 3 contains the computational aspects of Chapter 2. The core of this part consists in providing a way to obtain, on the computational level, unisolvence and minimality of a set of weights. This is, in general, done by looking at the size and the rank of the generalised Vandermonde matrix, which is first formalised and studied. Here linear algebra plays an interesting role, once one recognises that such am a t r i xc a nb et h o u g h to fa st h em a t r i xr e p r e s e n t i n gab i l i n e a rf o r mt h a tp a i r s small simplices and trimmed polynomial differential forms. An interesting formula which allows for computing its elements without quadrature formulas is recalled and discussed. Numerical tests to deduce its invertibility are offered and used to dwell about unisolvence and minimality. Since the inverse of the generalised Vandermonde matrix plays a central role in the subsequent, I also study its condition number, as well as a strategy to reduce it. In particular, it will be shown that well known coefficients, which make the monomial basis into the Bernstein one, improve the desired results. Again, numerical results that support this claim are provided. In the second section of this chapter the generalised Lebesgue constant is introduced, starting from the classical one. Its independence from the basis chosen for the polynomial space is proved. Results on the dependence of this quantity on the domain, and relative bounds, are computed. In the second part of this chapter I present a strategy to move from uniform to nonuniform simplices. The role of simplicial isomorphisms is underlined and I use such an idea to construct new families of small simplices from known ones. In particular, I provide a step by step construction based on points considered in nodal interpolation to extend the case of [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF] to simplices. I thus study in details construction related to warping and blending. As a consequence of these modifications, it turns out that proofs offered in Chapter 2 do not apply to the majority of these cases. In particular, while minimality is granted by construction, we see that it is not possible to offer a direct proof of unisolvence and results of the beginning of this chapter are invoked to solve this issue.

Chapter 4 contains numerical results that confirm the theory developed in Chapter 2 and Chapter 3. I first stress the role of the generalised Lebesgue constant xi in the context of simplicial interpolation,i . e . i n t e r po l a t i o no fd i ff e r e n t i a lf o r m sb y means of integration on simplices. To do so, I define formally an interpolator Π, based on weights, from smooth k-forms Λ k (T )t ot r i m m e dpo l y n o m i a ld i ff e r e n t i a l k-forms. I relate it to quantities previously defined and show some of its features, such as its commutativity in some particular cases. This functional is necessary, among others, to treat the following problem. Suppose that ! is the quantity one wants to measure in a physical experiment. Clearly one is only able to obtain e !,ap o s s i b l yp e r t u r b e dm e a s u r eo f!, due to some (unpredictable) errors. The generalised Lebesgue constant predicts the propagation of the error to Π! and Πe !. It is then crucial to obtain a reliable value for the generalised Lebesgue constant and I hence provide a strategy to estimate generalised Lebesgue constants associated with the desired sets of weights. I then use this technique to compare different choices of small simplices. As expected, the non uniform case offers more encouraging results in the above sense. A discussion on the subtlety of the concept of non uniformity in greater dimensions is pointed out. In fact, in order to extend distribution of nodes from edges to the interior of the simplex, a strong condition on rotational symmetry is imposed. A large amount of Tables, Figures and comparisons based on Matlab simulations is presented. I close this chapter studying if the techniques here presented and introduced may lead to a Runge counterexample in the context of interpolation of differential forms. The role of the generalised Lebesgue constant in this kind of problems is known [START_REF] Alonso Rodríguez | On a generalization of the Lebesgue's constant[END_REF]. In fact, the bigger the Lebesgue constant is, the bigger the (local) interpolation error may grow. Exploiting the already mentioned commutativity of the diagram, I am in a position to exhibit an explicit example in which a 1-form presents a Runge-like behaviour when the interpolator is associated with uniform simplices whereas it does not if one moves towards weights associated with non uniform distributions.

At the end of the thesis conclusions are gathered and results commented. I strongly put the accent on the necessity of the search for optimal simplices. xii Chapter 1

Preliminaries

This chapter is mainly divided into two parts. In the first, Notation,w eg a t h e r the notation that will be mostly used throughout the thesis. The second, which consists of the resulting three sections, is devoted to the introduction of basic and known concepts that we shall invoke in the rest of the thesis.

Notation

We mainly work with the n-dimensional Euclidean space R n and we shall present several examples in both R 2 and R 3 . When we do not assume the generality of R n we will stress the space in consideration. Elements of R n such as vectors and points are denoted with boldface symbols. To be more definite, we shall use Greek letters for vectors, such as ξ and η,a n di t a l i cl e t t e r sf o rp o i n t s ,m o s t l yx and y. We will also frequently identify a p oint x with the vector x 0. As usual, e i denotes the i-th element of the canonical basis of R n ,whic hisofteniden tifiedwith the corresponding point. We adopt the same conventions for the tangent space to R n at a point x,w h i c hi sd e n o t e db yt x R n ,s i n c et

x R n ⇠ = R n .
Similarly multi-indices, which we shall frequently think of as vectors, are denoted with boldface Greek symbols as well, typically α or θ.T h e weight of a multi-index α,w h i c hi st h es u mo fi t se l e m e n t s ,i sd e n o t e db yv e r t i c a lb a r sa si n |α| . = P i ↵ i . Permutations are usually denoted by and shall be thought of as maps k ! n, being in general `. = {0, 1,...,`}.B yp e r m u t a t i o nw ei nf a c tm e a na n y( i n c r e a s i n g ) injective function from a subset of the natural numbers N to another (possibly different) subset of the natural numbers N.T h es e to fp e r m u t a t i o n so fk elements out of n is denoted by S k n ,a n dw h e nk = n we write, for short, S n .W e w i l l consider only increasing permutations,i . e . pe r m u t a t i o n ss u c ht h a t (i) > (j)f o r i>j,u n l e s sd i ff e r e n t l ys pe c i fi e d . T h em a i ne x c e p t i o ni sL e m m a1 . 1 4 . Matrices are denoted by capital letters, for instance M or P .T h el e t t e rV is mostly reserved for the Vandermonde matrix. Notations M T and M T denote, respectively, the transpose and the inverse transpose of M .V e c t o rs p a c e so fm atrices, such as the space of n ⇥ n real matrices M n,n or the space of n ⇥ n invertible real matrices GL(R,n), follow the standard notation. The condition number of a matrix M is either denoted by cond(M )or 2 (M )whenw ew an ttostressthatw e are considering the 2-conditioning.

Simplices are denoted with capital letters and, in general, T is devoted to a generic n-simplex and F to its k-faces. Since the term face is often used to refer to (n 1)-faces of T ,incaseofam biguit ythedimensionofF is stressed. We shall also make use of the letter E in this context, especially when we deal with edges of T . When we want to invoke the k-simplex F associated with the permutation we use subscripts, e. g. F ,t os t r e s st h i sd e p e n d e n c e . T ot h eb o u n d a r yo fas i m p l e x F we reserve the notation @F,w h i c hi sc o n s i s t e n tw i t ht h a to ft h eo pe r a t o r@.

To small simplices, which we will formally define in Chapter 2, we dedicate the small italic letter s. Since they will frequently be characterised in terms of faces of T and permutations, subscripts may be placed to denote a specific small simplex. For instance we will encounter notations like s (Fσ,α) .S e t so fs m a l ls i m p l i c e sa r e denoted with capital letters, either Greek (mostly Σ) or italic (S and X in particular). Subscripts will denote the degree and more specific features of that set and superscript their (topological) dimension; between parentheses, the support of the small simplices is specified. As an example, Σ k r,min (T )d e n o t e st h es p a c eo fs m a l l k-simplices s 2 Σo fd e g r e er with some property ("min") that are supported (i.e. topologically contained) in T . Differential forms will be denoted by small Greek letters. In particular, to those objects is dedicated the letter ! and its variants like ! 0 ,etc. Thespacetheybelong to is properly stressed: Λ k denotes the space of smooth differential k-forms and appropriate condition on coefficients will be placed in front of that symbol. For instance, P r Λ k (T )d e n o t e st h es p a c eo fk-forms supported in T whose coefficient, in an appropriate chart, belong to the space P r of polynomial of degree less or equal than r.

Other mathematical symbols are standard: a hashtag # denotes the cardinality of a set. More peculiar notation and specific cases will be stressed and motivated at each proper time.

A crucial aspect of this work is the distinction between unisolvent and minimal degrees of freedom. We call unisolvent acollectionoflinearfunctionals{Φ i } i=1,...,N on a finite dimensional space P such that Φ i (p) = 0 implies that p =0f o re a c h p 2P, thus including the situation N dim P.I f ,i na d d i t i o n ,N =d i mP,w e call the collection {Φ i } i=1,...,N a (unisolvent and) minimal set of degrees of freedom for P.T h i sw i l lb ee x t e n s i v e l yt r e a t e di nC h a p t e r2 .

Simplices and complexes

The following is a standard introduction to discrete geometry. For complete proofs, technicalities and more specific details we refer to [START_REF] Hatcher | Algebraic topology[END_REF]. The general setting we work in is the Euclidean n-dimensional space R n .C o n s i d e rn +1 points {x 0 ,...,x n } in R n .W es h a l la l w a y st h i n ko fp o i n t sx i as column vectors, unless differently specified. Points {x 0 ,...,x k } are said to be affinely independent or in general position if, fixed one, say x 0 ,t h er e s u l t i n gv e c t o r sv i = x i x 0 are linearly independent, that is, if the matrix

M = v 1 | ...|v k 2M n,k is full rank. When k = n,inparticular,M 2 GL(R,n
). We will always work under this hypothesis. Note that any set of k +1 points extracted from a set of n +1 affinely independent points contains affinely independent points itself, since any submatrix of M of size n ⇥ k has rank k,be i n gi t sc o l u m n sl i n e a r l yi n d e pe n d e n t .

The convex hull of {x 0 ,...,x n } is the set of points x = P n i=0 i x i 2 R n with the constraints that i 0fori =0,...,n and P n i=0 i =1. Thesearesometimes called the affine combinations of {x 0 ,...,x n } and describe in fact the convex set generated by these points.

Any subset of k +1 elements of {x 0 ,...,x n } is described by a unique increasing permutation : k . = {0, 1,...,k}!n . = {0, 1,...,n}.F o r i n s t a n c e , t h e s e t {x 0 ,...,x n } is represented by the identity =i d:n ! n,t h es e tt h a tc o n t a i n s only one point {x i } is (0) = i and we associate the set containing, for example, {x 2 , x 4 , x 7 } with (0, 1, 2) = (2,4,[START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]. Thus any : k ! n represents one and a unique subset of {x 0 ,...,x n }.

Definition 1.1 (Simplex). Let {x 0 ,...,x n } be n +1 affinely independent points in R n .A nn-simplex T =[x 0 ,...,x n ] ⇢ R n
is the convex hull of n +1 affinely independent points; {x 0 ,...,x n } are called its vertices. A k-face (also called a k-subsimplex) F of T is the convex hull of k +1 vertices that generate T , namely

F =[x (0) ,...,x (k) ] ⇢ R n for some : k ! n.
As their name suggest, simplices1 are the simplest non trivial objects that one may construct with affinely independent points, see Figure 1.1. For this reason they play a crucial role in topology [START_REF] Munkres | Elements of Algebraic Topology[END_REF], homology theory [START_REF]l .8 1o fG r a d u a t eS t u d i e si nM a t h ematics[END_REF], algebraic topology [START_REF] Prasolov | Elements of Combinatorial and Differential Topology[END_REF] and many others disciplines.

x 0 x 1 x 2 x 2 x 0 x 1 x 3 Figure 1.1: A 2-simplex
is a triangle and a 3-simplex is a tetrahedron.

Since an (n 1)-face is determined by all the vertices of T but one, say the i-th, it is convenient to work under the convention that F i . =[ x 0 ,...,b x i ,...,x n ] denotes the (n 1) subsimplex of T generated by all the vertices of the simplex T with the exception of x i .T ot h i sw er e s e r v et h eh a tn o t a t i o n ,w h i c hm e a n st h a t x i has been removed.

The concept of "being a k-face" is an order relation on the collection of subsimplices which makes it a partially ordered set. Since this has a topological counterpart (if F 0 is a subsimplex of F then F 0 is contained in F ), we write it using the symbol of inclusion ✓,e . g . F 0 ✓ F .

The k-skeleton of an n-simplex T is the collection of its k-subsimplices and is denoted by ∆ k (T ). The number of k-subsimplices of an n-simplex coincides with the choice of k +1 v ertices out of n +1, and th us is #∆ k (T )= n+1 k+1 . Notice that ∆ n (T )=T and ∆ 0 (T )={x 0 ,...,x n }.R e a lv e c t o rs p a c e so fk-chains are constructed upon k-subsimplices ∆ k (T ). Definition 1.2 (Real k-chain). Let T be an n-simplex and ∆ k (T ) be its k-skeleton. A real k-chain is a formal linear combination of elements of ∆ k (T ), namely

c . = X F 2∆ k (T ) ↵ F F, with ↵ F 2 R and F 2 ∆ k (T ).
The associated real vector space is denoted by C k (T ).

An explicit relationship between k-and(k 1)-chains is provided by the boundary operator

@ : C k (T ) !C k 1 (T ) [x (0) ,...,x (k) ] 7 ! n X i=0 ( 1) i [x (0) ,...,b x (i) ,...,x (k) ], (1.1)
the hat indicating that x (i) has been removed. The representation in terms of increasing permutations : k ! n induces compatible orientations on T and its subsimplices with respect to the boundary operator @. It is a matter of untangling (1.1) to see that for any F 2 ∆ k (T ), for k =0 ,...,n,o n eh a s@(@F)=0 . T h i s fact is called the fundamental Lemma of homology. Acollectionofsimpliceswithappropriategluingconditioniscalledasimplicial complex.

Definition 1.3 (Simplicial complex). A simplicial complex X is a collection of simplices such that s 2 X and s 0 ✓ s implies that s 0 2 X and s, s 0 2 X implies that either s \ s 0 = ; or s \ s 0 is a face of both s and s 0 (i.e. s \ s 0 2 X). We shall always assume that such a collection is finite.

The k-skeleton of a simplicial complex is denoted by ∆ k (X) and contains all the k-simplices of X. The 0-skeleton of X is the vertex set vert(X) of X.

Intuitively, Definition 1.3 ensures closedness under the order relation ✓ and the operation of intersection. It is straightforward to observe that a single simplex is a simplicial complex itself and that, in such a case, the concepts of k-skeleton and vertex set boil down to those introduced for simplices. Figure 1.2: Left: a simplicial complex consisting of a 3-simplex (tetrahedron), five 2-simplices (triangles), ten 1-simplices (edges) and seven 0-simplices (points). Right: not a simplicial complex, since red and blue triangle do not intersect in a common face.

Since complexes are made of simplices and gluing conditions are provided, Definition 1.2 extends by linearity to the case of simplicial complexes and consequently does the boundary operator (1.1). Under the hypothesis of finiteness, the operator @ is thus represented by a rectangular matrix that contains only { 1, 0, 1}:o n c e an ordering for the k-a n dt h e( k 1)-simplices of X is chosen, the element in position i, j is equal to 0 if the j-th (k 1)-simplex is not part of the boundary of the i-th k-simplex, and ±1i fi ti s( s e e [ 5 8 ]o r[ 6 0 ]f o ra ne x p l i c i tc o m p u t a t i o n ) . The sign depends on the orientation and sometimes coefficients in Z 2 are invoked to avoid signs. This extends the concept of incidence matrix of a graph (see, for instance, the treatise [START_REF] Edelsbrunner | Computational topology, an introduction[END_REF]).

Barycentric coordinates

We have already observed that any p oint x 2 T is uniquely characterised by n +1 coefficients { 0 ,..., n } such that

x = n X i=0 0 x i , (1.2)
where the uniqueness of the representation is a consequence of belonging to a convex hull. Coefficients { 0 ,..., n } are called barycentric coordinates for T and may be extended to the whole R n if we drop the assumption that i 0f o r i =0,...,n. Now, define the matrix

A = ✓ x 0 x 1 ... x n 11... 1 ◆ .
Since {x 0 ,...,x n } are in general position, A is a square, invertible matrix. Considering (1.2) and the constraint P n i=0 i =1,onehasthatbarycen triccoordinates λ =( 0 ,..., n )o fa n yp o i n tx 2 T are readily computed as the unique solution of the linear system Aλ T =(x, 1) T ,n a m e l y

λ = A 1 ✓ x 1 ◆ .

Maps between complexes

Maps between simplicial complexes are called simplicial maps [START_REF] Munkres | Elements of Algebraic Topology[END_REF] and play the role of continuous maps in topology. In order to be consistent with Definition 1.3, some conditions should be imposed on mappings ' relating two complexes X and Y.

Definition 1.4 (Simplicial map). Let X and Y be simplicial complexes with vertex sets vert(X)={x 0 ,...,x n } and vert(Y)={y 0 ,...,y m }. A vertex map is a map

:v e r t ( X) ! vert(Y). Suppose that {x (0) ,...,x (k) } span a k-simplex of X if and only if their images { (x (0) ),..., (x (k) )} = {y (0) ,...,y (`) } span an `-simplex of Y, then in- duces a mapping ' : X ! Y.
This is called a simplicial map.

Chapter 1

The motivation of this definition is the following. Consider |X|,t h eunderlying space or body of X,w h ic histh etopologic als p ac egiv e nb yX embedded in R n and endowed with the induced topology. The vertex map :v e r t ( X) ! vert(Y)c a n be extended to a continuous map ' : |X|!|Y|.I np a r t i c u l a r ,s i n c e is a vertex map, its action on vert(X)i sw e l ld e fi n e d . W o r k i n gs i m p l e x -w i s ew et h u sd e fi n e

x = n X i=0 i (x)x i 7 ! '(x)= n X i=0 i (x) (x i ).
(1.3)

If x 2 X belongs to more than one simplex of X,t h ed e fi n i t i o no fs i m p l i c i a l complex ensures that the above mapping is well posed. Since {x (0) ,...,x (k) } span a simplex of X if and only if { (x (0) ),..., (x (k) )} span a simplex of Y,the mapping (1.3) is understood simplex by simplex and then extended to the whole complex X. This map is also continuous as a map '

: |X|!|Y|,a ss h o w ni n[ 5 0 , Lemma 2.3].
Notice that if we have extended the barycentric coordinates of each simplex s 2 X by 0 outside s then simplicial maps are meaningful only as mappings from s =[x (0) ,...,x (k) ]t o'(s)=[ (x (0) ),..., (x (k) )]; in case of other extensions of barycentric coordinates restrictions should be invoked.

The definition of simplicial map may seem restrictive, but in fact it is not. Consider, for instance, the complex given only by the 3-simplex T =[x 0 , x 1 , x 2 , x 3 ] and the complex given only by the 2-simplex E =[x 0 , x 1 ]. The map This lemma avoids situations similar to that previously described, in which the complex obtained under the action of the simplicial map has few to do with the domain of '. We thus have the following definition (see also [START_REF] Edelsbrunner | Computational topology, an introduction[END_REF]Chapter 3]). Definition 1.6 (Simplicial isomorphism). Let X and Y be simplicial complexes. Let :v e r t ( X) ! vert(Y) be a bijective vertex map such that {x (0) ,...,x (k) } span a k-simplex of X if and only if { (x (0) ),..., (x (k) )} span a k-simplex of Y. The induced simplicial map ' is said to be a simplicial isomorphism.

: {x 0 , x 1 , x 2 , x 3 }!{x 0 , x 1 } such that (x 0 )= (x 2 )= (x 3 )=x 0 , (x 1 )=x 1 ,i
Loosely speaking, simplicial isomorphisms map k-simplices to k-simplices preserving adjacencies and avoiding the generation of hanging nodes. For this reason we shall extensively use them in Chapter 3.

Differential forms on simplices

Let Ω ⇢ R n .A( s m o o t hdifferential ) k-form on Ω is a smooth section of the k-th exterior power of the cotangent bundle of Ω,

! :Ω ! ^k(tΩ) ⇤ , being tΩ . = S x2Ω t
x Ω the tangent bundle of Ω, that is, the collection of the tangent spaces to Ω at any x 2 Ω. This definition extends to smooth manifolds by means of (local) charts U ✓ Ω ! R n and to manifold with corners, such as simplices T , as well by considering charts U ✓ T !{ x 2 R n : x 1 0,...,x k 0}.W ew i l l mainly work in this framework. We leave technical details to [START_REF] Lee | o l .2 1 8o fG r a d u a t eT e x t si n Mathematics[END_REF] and we address to [START_REF] Christiansen | Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension[END_REF] for a more finite element oriented approach. Note that, by definition, 0-forms Λ 0 (T ) are just smooth functions T ! R and their integral over a point assumes the meaning of evaluation. For each

! 2 Λ 0 (T )a n dx 2 T ,o n ei nf a c th a s Z x ! = !(x).
Differential forms in R n are globally defined, since the tangent space at any point x 2 R n is t x R n ⇠ = R n ,a n ds ot h e ya r eo nT ⇢ R n . Hence, if {x 1 ,...,x n } are coordinates for R n ,ak-form may be uniquely written in the multi-index notation dx

I . =dx i 1 ^...^dx i k for I =(i 1 ,...,i k )a s ! = X |I|=k f I (x 1 ,...,x n )dx I , (1.4)
where each f I : R n ! R is a smooth function and {dx 1 ,...,dx n } is a basis for the cotangent space, dual to the basis n @ @x 1 ,..., @ @xn o of the tangent space t x R n . Differential forms may also be characterised in terms of multilinear alternating tensors, which means that for

! 2 Λ k (T ), a, b 2 R and v 1 ,...,v k 2 R n ,o n eh a s !(v 1 ,...,av i + bv j ,...,v k )=a!(v 1 ,...,v i ,...,v k )+b!(v 1 ,...,v j ,...,v k ) and !(v 1 ,...,v i , v j ,...,v k )= !(v 1 ,...,v j , v i ,...,v k ).
In particular, if v i = v j for some i 6 = j, !(v 1 ,...,v i ,...,v j ,...,v k ) = 0. This basic fact will be extensively used in what follows.

Chapter 1

Being defined as an exterior algebra, differential forms are naturally endowed with the wedge product.I ti sam a p p i n g

^:Λ k (T ) ⇥ Λ `(T ) ! Λ k+`( T ) (!, ⌘) 7 ! ! ^⌘,
whose properties are a direct consequence of the interpretation as alternating tensors. Also, since differential forms are alternating we have dx ^dy = dy ^dx. This implies that dx ^dx = 0. In particular, with respect to the above equation, when k + `>n, ! ^⌘ =0.

The exterior derivative or differential di sal i n e a ro pe r a t o r

d: Λ k (T ) ! Λ k+1 (T ) that acts on ! = f (x 1 ,...,x n )dx I 2 Λ k (T )a s ! 7 ! d! . = n X i=1 @f(x 1 ,...,x n ) @x i dx i ^dx I
and is extended to a generic element of Λ k (T ) by linearity. To be more definite, writing in some charts ! = P |I|=k f I (x 1 ,...,x n )dx I ,i t sd i ff e r e n t i a lr e a d s

d! = X |I|=k n X j=1 @f I (x 1 ,...,x n ) @x j dx j ^dx I .
Two fundamental (vector) subspaces of Λ k (T )aredefinedintermsofd. Thespace of closed forms,w h i c hc o n t a i n s! 2 Λ k (T )s u c ht h a td ! =0 ,a n dt h a to fexact forms.W es a yt h a tak-form ! is exact if there exists a form ⌘ 2 Λ k 1 such that ! =d⌘.S u c ha n⌘ is said to be a potential for !.

The fact that differential forms are alternating and Schwartz's Theorem, which states that any smooth function f (x 1 ,...,x n )o be y s

@ 2 f (x 1 ,...,x n ) @x i @x j = @ 2 f (x 1 ,...,x n ) @x j @x i ,
yield the following result, which is known as the fundamental Lemma of cohomology.

Lemma 1.7. Let ! 2 Λ k (T ). Then d(d!)=d 2 ! =0.
This Lemma shows that an exact form is closed, whereas the vice versa is not granted. This opens the way to cohomology theory [START_REF] Bott | Differential forms in algebraic topology,v o l .8 2o f Graduate Texts in Mathematics[END_REF].

In Section 1.1 we have introduced change of coordinates form Cartesian to barycentric. In terms of differential forms this is a special case of pullback.S u p p o s e T and T 0 are n-simplices and ' : T ! T 0 is a sufficiently regular bijection. Then its differential d' is a linear mapping between the respective tangent bundles, called pushforward,r e p r e s e n t e di ne a c hp o i n tx 2 T by a n ⇥ n matrix d' x . It thus induces a contravariant map called pullback

' ⇤ :Λ k (T 0 ) ! Λ k (T ) ! 7 ! ' ⇤ ! such that ' ⇤ !(v 1 ,...,v k ) . = ! (d'(v 1 ),...,d'(v k ))
for any choice of k vector fields {v 1 ,...,v k } of (the tangent bundle of) T .I n coordinates, if {x 1 ,...,x n } are coordinates for T and {y 1 ,...,y n } are coordinates for T 0 ,t h ep u l l b a c ko f! = f (y 1 ,...,y n )dy I reads as

' ⇤ ! = f (' 1 (x 1 ,...,x n ),...,' n (x 1 ,...,x n )) 0 @ n X j=1 (@' i 1 ) j @x j dx j 1 A ^...^0 @ n X j=1 (@' i k ) j @x j dx j 1 A .
Pullback yields a generalisation of the change of variables in integration Z

'(T ) ! = Z T ' ⇤ !. (1.5)
Af u n d a m e n t a lr e s u l to ni n t e g r a t i o no fk-forms relates the exterior derivative d and the boundary operator @. It is Stokes' Theorem and we here report its version for simplices, which can be found in [START_REF] Lee | o l .2 1 8o fG r a d u a t eT e x t si n Mathematics[END_REF]Theorem 16.25].

Theorem 1.8 (Stokes' Theorem for simplices). Let T be a k-simplex and let ! 2

Λ k 1 (T ). Then Z T d! = Z @T !.
Ap a r a l l e lc o n s t r u c t i o nw i t hc o e ffi c i e n t si nS o b o l e vs p a c e si sp o s s i b l ea n dh a s been carried in [START_REF]Sobolev spaces of differential forms and de Rham-Hodge isomorphism[END_REF].

A norm for differential forms

Integration of k-forms easily offers the introduction of an L 2 -norm for Λ k (T ). It is defined by means of the Hodge star operator

? :Λ k (T ) ! Λ n k (T ), (1.6)
Chapter 1

which is a linear isomorphism between k and (n k)-forms. For a formal definition we refer to [START_REF] Warner | Foundations of Differentiable Manifolds and Lie Groups[END_REF]. For our needing, we shall only use the fact that it reverses the alternating part dx I of ! adjusting the sign, so that ?dx I =d x J such that dx

I ^dx J =dx 1 ^...^dx n .T h u st h eL 2 -norm for Λ k (T )i s k!k 2 . = ✓Z T ! ^?! ◆ 1 2
.

(1.7)

Note that when ! 2 Λ 0 (T ), i.e. when ! is a smooth function ! : T ! R,thisnorm reduces to the usual L 2 -norm for square integrable functions. In fact, in such a case ?! = !dx 1 ^...^dx n and hence one retrieves

k!k 2 = R T |!| 2 dx 1 ^...^dx n 1 2
. This norm is not particular informative in terms of k-forms, since it hinges on integration on the whole n-simplex T of the n-form ! ^?!,a n dt h u sy i e l d sglobal information on T .T oo b t a i nm o r elocal features another norm is required. Such an o r mi sk n o w na st h e0 -norm and was studied in [START_REF] Harrison | Continuity of the integral as a function of the domain[END_REF]. It is defined as

k!k 0 . =sup c 1 |c| 0 Z c ! . (1.8)
Here c is supposed to range over all the possible real simplicial k-chains supported in T ,a sf o r m a l i s e di n [START_REF] Alonso Rodríguez | On a generalization of the Lebesgue's constant[END_REF] ,a n d|c| 0 denotes its mass, which is its k-volume R c dvol. We assume that the mass of a point is 1. Equation (1.8) defines a norm [START_REF] Harrison | Continuity of the integral as a function of the domain[END_REF].

Note that for k =0,thatis,forfunctions,norm(1.8)reducestotheusualsup norm (or 1-norm) for functions. In fact, in this case ! is a smooth function and

|c| 0 =1. Expandingw eget k!k 0 =sup c 1 |c| 0 Z c ! =sup c Z c ! =sup c |!(c)| .
As a consequence, the 0-norm does not make Λ k (T )aBanachspaceevenfork =0.

High order Whitney forms

There are two ways of defining the spaces P r Λ k (T )o fhigh order Whitney forms. The first, which is intrinsic and algebraic, is based on the Koszul operator (that is, interior multiplication on differential forms), and is carried in [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF] and well summarised in [START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF]. The second one, which is purely geometrical, hinges on the construction of degree one Whitney forms offered in [START_REF] Whitney | Geometric Integration Theory[END_REF] and then is carried by multiplying such forms for appropriate polynomials, as in [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF]. We present them both as we shall borrow peculiarities from both the two approaches.

Whitney forms via Koszul operator

The space Λ k (R n )o fs m o o t hk-forms admits subspaces that can be described in terms of the coefficients of !. In particular, consider the space of polynomials of Chapter 1

Weights as dofs for high order Whitney forms degree r on R n ,w h i c hw ed e n o t eb yP r (R n ). We have an obvious decomposition into spaces of homogeneous polynomials of n variables as

P r (R n )= r M i=0 H i (R n ).
Dropping the dependence, we denote by P r Λ k (R n )t h es p a c eo fd i ff e r e n t i a lf o r m s whose coefficients belong to P r (R n ), i.e. that are polynomials of degree r in n variables. Of course, one also has

P r Λ k (R n )= r M i=0 H i Λ k (R n ).
One then has

dim P r Λ k (R n )=dimP r (R n )dimΛ k (R n )= ✓ n + r r ◆✓ n k ◆ . (1.9)
Since there is an embedding i : T, ! R n , we may define

P r Λ k (T ) . = i ⇤ P r Λ k (R n ) ⇢ Λ k (T ). (1.10)
Clearly, the dimension of P r Λ k (T )c o i n c i d e sw i t ht h a to fP r Λ k (R n ). We shall lo ok for subspaces of P r Λ k (R n )andth usofP r Λ k (T ). To this end, we introduce the Koszul differential. Fix a vector v 2 R n .T h einterior multiplication [START_REF] Lee | o l .2 1 8o fG r a d u a t eT e x t si n Mathematics[END_REF]Chapter 14] with v is defined as the contraction

 v :Λ k (R n ) ! Λ k 1 (R n )  v !(v 1 ,...,v k 1 )=!(v, v 1 ,...,v k 1 ), (1.11)
for all v 1 ,...,v k 1 2 R n .S i n c e ! is alternating, one has that  v ( v !)=0 and for this reason the operator  v is referred to as Koszul differential. We shall always assume that v is the identity vector field, i.e. the vector that maps at each point x 2 R n the origin 0 2 t 0 R n to x 2 t x R n ⇠ = R n ,s ot h a tv = x 0 = x. Under this assumption, in coordinates the interior mutliplication acts on ! = p(x 1 ,...,x n )dx (1) ^...^dx (k) as

 v ! = k X i=1 ( 1) i p(x 1 ,...,x n )x (i) dx (1) ^...^c dx (i) ^...^dx (k) , (1.12)
the hat meaning that dx (i) has been removed, and extends by linearity. By restricting to T we thus get an operator  v on P r Λ k (T ). Observe that the action of  v preserves homogeneous polynomial differential forms and hence it might be read as a mapping

 v : H r Λ k (T ) !H r+1 Λ k 1 (T )
Chapter 1 that increases by 1 the polynomial degree while it decreases the order of the form by 1.

We are in a position to define the spaces of high order Whitney forms,s o m etimes called also spaces of trimmed polynomial differential forms [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]. Definition 1.9. We put

P r Λ k (T ) . = ! 2P r Λ k (T ): v ! 2P r Λ k 1 (T ) . (1.13)
The space P r Λ k (T )i sap r o p e rs u b s p a c eo fP r Λ k (T )a n ds t r i c t l yc o n t a i n s P r 1 Λ k (T ), as the following easy example shows.

Example 1.10. The 1-form ! = x 1 dx 1 belongs to

P 1 Λ 1 (R n ) but not to P 1 Λ 1 (R n ), since  v ! = x 2 1 2P 2 Λ 0 (R n ). The 1-form ! = x 1 dx 2 + x 2 dx 1 belongs to P 1 Λ 1 (R n ) since  v ! =02P 1 Λ 0 (R n ) but not to P 0 Λ 0 (R n ).
More generally, one has inclusions

P r 1 Λ k (T ) ✓P r Λ k (T ) ✓P r Λ k (T ),
where the first equality holds for k = n and the second for k =0,asaconsequence of (1.11).

Definition 1.9 is difficult to figure out but crucial to establish algebraic features of P r Λ k (T ), such as its dimension. To begin with, observe that there is a direct sum decomposition

P r Λ k (T )=P r 1 Λ k (T )  v H r 1 Λ k+1 (T ), (1.14)
which is a consequence of (1.12). Note that we may also write

P r Λ k (T )=P r 1 Λ k (T )+ v P r 1 Λ k+1 (T ). (1.15)
Exploiting this fact, one computes (see [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]Equation (3.15)])

dim P r Λ k (T )= ✓ r + k 1 k ◆✓ n + r n k ◆ . (1.16)
We gather in the following proposition several facts that we are going to exploit frequently in subsequent chapters. Proposition 1.11 encloses Theorem 5.1a n d Theorem 5.4o f [START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF]a n dt h em a i nr e s u l t so fS e c t i o n3 .2o f [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF] . F o re a s eo ft h e reader, we rewrite a detailed proof.

Proposition 1.11. The following results hold.

(i) Let ⌘ 2H r Λ k (T ). One has

(d v +  v d)⌘ =(r + k)⌘.
(1.17)

(ii) Let ! 2P r Λ k (T ). One has (ii.a) if d v ! =0, then  v ! =0; (ii.b) if  v d! =0, then d! =0. (iii) If ! 2P r Λ k (T ) is such that d! =0, then ! 2P r 1 Λ k (T ).
(iv) Let T 0 be another n-simplex and let ' : T 0 ! T be an affinity that maps T 0 to T . Then

' ⇤ P r Λ k (T ) = P r Λ k (T 0 ). ( 1
Λ k (R n )= L r i=0 H i Λ k (R n
) and prove the fact for ⌘ 2H r Λ k (T ), the result will follow. Since the operator  v is linear, for any c 2 R one has c v ⌘ =  v c⌘,s o v ⌘ =0i fa n do n l yi f v c⌘ =0 . In particular, take c =(r + k)a n dc o m p u t e

 v (r + k)⌘ =  v (d v +  v d)⌘ =  v d v ⌘, since  v ( v !)=0 . S i n c ed  v =0b yh y p o t h e s i sw eg e t v (r + k)⌘ =0 ,t h u s  v ⌘ =0.
Likewise, we check (ii.b) as

d(r + k)⌘ =d(d v +  v d)⌘ =d v d⌘, since d 2 =0
,andw econcludeinthesamew a y . (iii).T os h o wt h a tac l o s e df o r mi nP r Λ k (T )b e l o n g si nf a c tt oP r 1 Λ k (T ) we use the decomposition (1.14). We split

! = ⇢ +  v ⌘ with ⇢ 2P r 1 Λ k (T )a n d ⌘ 2H r 1 Λ k+1 (T ). Computing the differential we get 0=d! =d⇢ +d v ⌘.
Observe that, since d v ⌘ 2H r 1 Λ k+1 (T )a n dd e gd ⇢  r 2, their sum vanishes if and only if both terms vanish. It follows that, in particular, d v ⌘ =0a n db y (ii)

 v ⌘ = 0. Hence ! = ⇢ 2P r 1 Λ k (T ).
(iv).W ep r o v et h a t( 1 . 1 8 )h o l d s .C o n s i d e rc o o r d i n a t e sy = {y 1 ,...,y n } on T 0 and x = {x 1 ,...,x n } on T and denote by ' : T 0 ! T the non degenerate affinity that maps T 0 to T ,w h i c ht r a n s f o r m sc o o r d i n a t e sa sy 7 ! '(y)=Ay + b = x, whence dx =d( Ax + b)=Ady.F o re a s ew ea r et h i n k i n go fx and y as column vectors. We compute the pullback ' ⇤ :Λ k (T ) ! Λ k (T 0 )byapplyingthedefinition. Consider ! = p(x)dx I = p(x)dx (1) ^...^dx (k) 2P r Λ k (T ). We have

' ⇤ ! =(p ')(y) n X j=1 A (1),j dy j ! ^...^ n X j=1 A (k),j dy j ! .
Now, (p ')(y)i si nf a c tap o l y n o m i a lo fd e g r e er in y being the composition of ap o l y n o m i a lo fd e g r e er and one of degree 1, thus ' ⇤ ! is the sum of polynomial differential forms of degree at most r. Notice that the same can be said if p 2 H r (T ), although in such a case (p ')(y) is not necessarily an element of H r (T 0 ) but belongs in general to P r (T 0 ). The result thus follows immediately applying the linearity of the pullback to the decomposition given in (1.15)

' ⇤ ! = ' ⇤ (⇢ +  v ⌘)=' ⇤ ⇢ + ' ⇤  v ⌘, being ' ⇤ ⇢ 2P r Λ k (T )a n d' ⇤  v ⌘ 2P r 1 Λ k+1 (T ).

Geometrical construction of Whitney forms

We may now turn to the geometrical construction of the spaces P r Λ k (T ). To begin with, we construct the space P 1 Λ k (T ). To this end, barycentric coordinates { 0 ,..., n } are considered on T .T h e c o n v e n i e n c e o f t h i s c h o i c e i s m o t i v a t e d in [START_REF] Gopalakrishnan | Nédélec spaces in affine coordinates[END_REF]. Note that by differentiating the constitutive relationship of barycentric coordinates P n i=0 i =1w eimmediatelygetthat

d i = X i6 =j d j ,
hence {d 0 ,...,d n } are linearly dependent.

In Section 1.1 we provided a bijection between increasing permutations : {0,...,k}!{ 0,...,n} and ∆ k (T ), the space of k-subsimplices of the n-simplex T .T os t r e s st h i sf a c t ,i nt h i sS e c t i o nw em a k ee x p l i c i tt h i sd e p e n d e n c e 7 ! F .

On every F 2 ∆ k (T )o n l yk +1 barycentric coordinates out of n +1 survive; in particular it follows from the structure of k-simplex of F that the only non vanishing are those in the range of .W ea s s o c i a t eap o l y n o m i a ld i ff e r e n t i a lkform with each F 2 ∆ k (T )b ys e t t i n g

! . = k X i=0 ( 1) i (i) d (0) ^...^c d (i) ^...^d (k) , (1.19)
the hat meaning that d (i) has been removed. The coefficient of this form is a degree 1 polynomial in the coordinates { 0 ,..., n }.T h eq u a n t i t y( 1 . 1 9 )i st h e Whitney form associated with F . To stress this dependence we shall frequently denote it by placing either the subscript or F next to the form !.S i n c ew e have bijections 7 ! F 7 ! ! we have n+1 k+1 Whitney k-forms on an n-simplex T .

x 2

x 0

x 1 Example 1.13. Let T ✓ R 2 be endowed with barycentric coordinates 0 , 1 , 2 . We assign to each F 2 ∆ 1 (T ) its Whitney form ! . There are three increasing permutations {0, 1}!{ 0, 1, 2}, that are {0, 1} 7 ! {0, 1}, {0, 1} 7 ! {1, 2} and {0, 1} 7 ! {0, 2}. Applying (1.19) we thus obtain three Whitney 1-forms:

x 3 ! = 2 ! = 0 d 2 2 d 0 ! = 0 d 1 ^d 3 1 d 0 ^d 3 + 3 d 0 ^d 1 ! T =dvol T =d 1 ^d 2 ^d 3
! 1 = 0 d 1 1 d 0 , ! 2 = 0 d 2 2 d 0 , ! 3 = 1 d 2 2 d 1 .
The following result, which is very close to [21, Proposition 3.2], enlightens the handy behaviour of Whitney forms on simplices, and the identities there contained will be extensively used throughout the whole work. We closely follow the already known proof. Lemma 1.14. Let T be a simplex and F 2 ∆ k (T ).I f! is the Whitney form associated with F , then

! Fσ =d (1) ^...^d (k) Fσ .
(1.20)

Moreover, Z

Fσ ! = 1 k! . (1.21)
It is worth noting that, in some contexts, such as in Whitney's original work [START_REF] Whitney | Geometric Integration Theory[END_REF] and some of its consequences such as [START_REF] Dodziuk | Finite-difference approach to the Hodge theory of harmonic forms[END_REF] and [START_REF] Dodziuk | Riemannian structures and triangulations of manifolds[END_REF] or the more recent work by Christiansen and Rapetti [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF], it is considered convenient to place a coefficient k! in front of the right hand side of (1.19). This makes some computations cleaner and, of course, some other heavier. For instance the coefficient k!m a k e s( 1 . 2 0 ) read as

! Fσ = k!d (1) ^...^d (k) Fσ ,butyieldsinturnanicerversionof(1.21), which is R Fσ ! =1.
Proof. Let S k denote the set of all permutations {0,...,k}!{0,...,k} (not only increasing). We first show that, letting ↵ vary in S k , we may write

! = 1 k! X ↵2S k sgn(↵) (↵(0)) c d (↵(0)) ^d (↵(1)) ^...^d (↵(k)) , (1.22)
where sgn(↵) denotes the sign of the permutation ↵ and makes all the summands in the above expression positive. For each i =0, 1,...,k,w efi xt h epe r m u t a t i o n s ↵ 2 S k such that ↵(0) = i (thus, we have (↵(0)) = (i) )a n dd e n o t eb yS k,i this set. We have that the cardinality of this set, i.e. the number of such permutations, is #S k,i = k!. We compute

1 k! X ↵2S k sgn(↵) (↵(0)) c d (↵(0)) ^d (↵(1)) ^...^d (↵(k)) = 1 k! k X i=0 X ↵2S k,i sgn(↵) (i) c d (↵(0)) ^d (↵(1)) ^...^d (↵(k)) = 1 k! k X i=0 (i) X ↵2S k,i sgn(↵)d (↵(1)) ^...^c d (↵(0)) ^...^d (↵(k)) .
Since is an increasing permutation, replacing ↵ and reordering, one has

sgn(↵)d (↵(1)) ^...^c d (↵(0)) ^...^d (↵(k)) =( 1) i d (0) ^...^c d (i) ^...^d (k) .
Plugging this in, we obtain

! = 1 k! k X i=0 ( 1) i (i) X ↵2S k,i d (
! Fσ = k! k! k X i=0 (i) ! d (1) ^...^d (k) Fσ =d (1) ^...^d (k) Fσ ,
which also yields

Z Fσ ! = Z Fσ d (1) ^...^d (k) = 1 k! .
The last equality can be deduced, in particular, observing that it represents the volume of the standard k-simplex.

As an immediate consequence of Lemma 1.14 we have the following.

Corollary 1.15. Let T be an n-simplex and F 2 ∆ k (T ). Let ! be the corresponding Whitney form. If F 0 2 ∆ k (T ) with 6 = 0 , ! F σ 0 =0and, as a consequence, Z

F σ 0 ! =0.
Proof. Since d is a linear operator, if i vanishes so does d i .S i n c eF 6 = F 0 , there exists i such that i =0o nF 0 and so d i F σ 0 =0 . S i n c et h er e s t r i c t i o ni s linear and distributes over wedge product, in view of (1.20) we get ! F σ 0 =0.

Using Lemma 1.14 one checks that all Whitney k-forms ! are linearly independent as F ranges in ∆ k (T ). This justifies the following definition. Definition 1.16 (Space of Whitney forms). Let T be an n-simplex. The space of Whitney k-forms is

W k (T ) . =span{! : F 2 ∆ k (T )} . (1.24)
By direct computation one checks that, for each

F 2 ∆ k (T ), the associated Whitney k-form satisfies  v ! 2P 1 Λ k 1 (T ).
Their number equals that of ksubsimplices of T ,w h i c hi s n+1 k+1 , and coincides with dim P 1 Λ k (T ). To see this, it suffices to apply formula (1.16) to r =1 . O n eh a si nf a c tt h ef o l l o w i n gr e s u l t , which is [7, Theorem 4.1].

Theorem 1.17. One has

W k (T )=P 1 Λ k (T ).
This result shows that Whitney k-forms associated with ∆ k (T )a r eab a s i sf o r P 1 Λ k (T ). In order to extend Definition 1.16 to a generic r>1o n es h a l li n v o k e again the Koszul differential, so we go back to Definition 1.9. Interestingly, as pointed out in [1], Whitney forms are linearly independent over R,b u tn o to v e r spaces of polynomials in general. This is easily seen from the fact that we have a surjection

P r 1 (T ) ⇥P 1 Λ k (T ) !P r Λ k (T ), but dim P r 1 (T ) ⇥ dim P 1 Λ k (T ) dim P r Λ k (T ),
with equality holding only for extreme cases. In other words, the product of a basis of P r 1 (T )w i t ho n eo fP 1 Λ k (T )g i v e sas y s t e mo fg e n e r a t o r sf o rP r Λ k (T ) but not a basis for such a space. Consider multi-

indices α =(↵ 0 ,...,↵ n )ofweight |α| . = P n i=0 ↵ i = r 1a n dp u tλ α . = Q n i=0 ↵ i i .
As pointed out in [1], it is possible to retrace [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]Section 4.4] to offer an algorithm that discards generators up to obtaining a basis. This procedure yields a monomial basis λ α ! , |α| = r 1, that can be de-homogenised by choosing a barycentric coordinate, say the 0-th, and substituting 0 =1

P n i=1 i . First, put I(n +1,r 1) the set of multiindices of n +1 variables and weight r 1. For what we have observed before, the set {λ α ! : α 2I(n +1,r 1),F 2 ∆ k (T )} spans P r Λ k (T )b u tb e a r ss o m e redundancy. For each F =[ x (0) ,...,x (k) ] 2 ∆ k (T )a n dr 1, we define the follwing subset of I(n +1,r 1):

I (n +1,r 1) . = {α 2I(n +1,r 1) : ↵ i =0for i< (0)} . (1.25)
Loosely speaking, since any F 2 ∆ k (T )isparametrisedbyanincreasingpermutation : {0,...,k}!{0,...,n},w ed i s c a r da l lt h em u l t i -i n d i c e sα =(↵ 0 ,...,↵ n ) that present a non-zero element before (0).

The following dimension count relates the collection of sets I (n +1,r 1) as

F varies in ∆ k (T )a n dP r Λ k (T ). Lemma 1.18. Let : k ! n be an increasing permutation such that (0) = `. One has #I (n +1,r 1) = ✓ n k ◆✓ n + r ` 1 r 1 ◆ . (1.26) As a consequence, if T is an n-simplex, one has X 2S n k #I (n +1,r 1) = dim P r Λ k (T ). (1.27)
Proof. First, recall that any F 2 ∆ k (T )i sr e p r e s e n t e db ya ni n c r e a s i n gpe r m u t ation : {0,...,k}!{0,...,n}.S u p p o s e (0) = `.I tf o l l o w st h a t{ (1),..., (k)} maps into the remaining n `terms of {`+1,...,n},h e n c eo n em a yt h i n ko fi t as a permutation in

S n ` 1 k 1 .T h u s #I (n +1,r 1) = ✓ n k ◆✓ n `+ r 1 r 1 ◆ and (1.26) is established.
To prove (1.27), we let vary in S n k and observe that 0  (0)  n k,be i n g increasing. Computing each term in the sum by (1.26), we get

X 2S n k #I (n +1,r 1) = n k X `=0 ✓ n k ◆✓ n + r ` 1 r 1 ◆ = ✓ r + k 1 k ◆✓ n + r n k ◆ =dimP r Λ k (T ),
see (1.16). The result is established.

Along with the above dimension count we get the following result, which is [1, Theorem 2] (see also references therein, such as [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]).

Theorem 1.19. The set

{λ α ! : F 2 ∆ k (T ), α 2I (n +1,r 1)} is a basis for P r Λ k (T ).
This characterisation allows to extend results for P 1 Λ k (T )t oP r Λ k (T ). One in fact has the following, which is the counterpart of Corollary 1.15 for k>1.

Lemma 1.20. Let F ,F 0 2 ∆ k (T ) and let ! = λ α ! 2P r Λ k (T ).I f 6 = 0 , one has ! F σ 0 =0. (1.28)
As a consequence, Z

F σ 0 ! =0. (1.29) Proof. One has ! F σ 0 = λ α F σ 0 ! F σ 0 =0,since! F σ 0 =0
,assho wninLemma1.14 and its corollary. One also gets that

R F σ 0 ! =0.
Degrees of freedom for P r Λ k (T )

The importance of (high order) Whitney forms has been recognised in several papers; in particular, their role in electromagnetism has been widely studied (see, for example, [START_REF] Bossavit | Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism[END_REF], [START_REF]Finite elements in computational electromagnetism[END_REF] or [START_REF] Monk | Finite Element Methods for Maxwell's Equations[END_REF]). In [START_REF] Hiptmair | Canonical construction of finite elements[END_REF] Hiptmair showed that in the case of R 3 they are a representation via differential forms of Nédélec first family of finite elements [START_REF] Nédélec | Mixed finite elements in R 3[END_REF]. Generalisations to R n , which justify the role of the spaces P r Λ k (T )h e r e introduced, have been offered in [START_REF] Gopalakrishnan | Nédélec spaces in affine coordinates[END_REF]. In this duality between differential forms and functions of several variables we call proxy fields the functional counterpart of differential forms. Most common degrees of freedom for P r Λ k (T )a r emoments,a l s oi n t r o d u c e d in [START_REF] Nédélec | Mixed finite elements in R 3[END_REF] and rewritten in terms of differential forms in [START_REF] Hiptmair | Canonical construction of finite elements[END_REF]. In [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF] and [START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF] they were also used to decompose very general spaces of polynomial differential forms and their dual, exploiting their relationship with Nédélec second family of finite elements [START_REF]A new family of mixed finite elements in R 3[END_REF]. We thus recall the key points of this construction, that correspond to [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]Section 4.6].

Definition 1.21 (Moments for ! 2 Λ k (T )). Let T be an n-simplex and ! 2 Λ k (T ). The quantity Z F ! F ^⌘ (1.30) is the moment of ! associated with F 2 ∆ k (T ) and ⌘ 2P r+k dim F 1 Λ dim F k (F ).
Chapter 1

Weights as dofs for high order Whitney forms Equation (1.30) defines thus linear functionals

! (F, ⌘) . = ⌘ 7 ! ✓ ! 7 ! Z F ! F ^⌘◆ (1.31)
and [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]Theorem 4.12] shows that for ! 2P r Λ k (T )

! (F, ⌘)=0 8⌘ 2P r+k dim F 1 Λ dim F k (F ),F2 ∆ k (T )
implies that ! =0. This,alongwithadimensioncoun t,whic his[7,Theorem4.13], makes moments of Definition 1.21 unisolvent and minimal degrees of freedom for P r Λ k (T )i ns u c has e n s e . T h i sc o n c e p tw i l lbee x p a n d e di nC h a p t e r2 . Note that, when r =1 ,m o m e n t sf o rP 1 Λ k (T )b o i ld o w nt oi n t e g r a l so nksubsimplices of T .T h i si saw a yt os h o wt h a ti n t e g r a t i o no n∆ k (T )y i e l d sd etermining degrees of freedom for degree 1 Whitney k-forms. These integrals are called weights and will be extensively studied in the sequel.

Proposition 1.22. Let ! 2P 1 Λ k (T ).I f Z F ! =0 8F 2 ∆ k (T ), then ! =0.
Proof. Following (1.24), write ! = P a ! ,w i t ha 2 R. Now, as a consequence of Corollary 1.15, we get Z

F σ 0 ! = Z F σ 0 X a ! = a 0 Z F 0 σ ! 0 = 1 k! a 0 ,
where last equality is (1.21). Thus R F σ 0 ! 0 =0i fa n do n l yi fa 0 =0 . S i n c et h i s holds for all the coefficients of ! = P a ! ,w ec o n c l u d e! =0.

This allows to construct an interpolator from Λ k (T )toP 1 Λ k (T )byassociating any ! 2 Λ k (T )w i t ht h eu n i q u ef o r mi nP 1 Λ k (T )t h a ts h a r e sd e g r e e so ff r e e d o m with !.S i n c em o m e n t sa n dw e i g h t sc o i n c i d ef o rr =1,w eunam biguouslyput

Π k T :Λ k (T ) !P 1 Λ k (T )( 1 . 3 2 ) ! 7 ! Π k T ! . = X F 2∆ k (T ) ✓Z F ! ◆ ! F .
In what follows we will also consider the restriction of this projector to the space P r Λ k (T ) in place of Λ k (T ), as done in [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF]. This does not require any new definition, since

P r Λ k (T ) ⇢ Λ k (T )
The main topic of this work is to extend the concept of weight and hence generalise Proposition 1.22 to P r Λ k (T ).

Chapter 2 Weights, unisolvence and minimality

Following the classical approach (see, for instance, [23], [START_REF]The finite element method for elliptic problems,vol.4ofStudiesinMathematics and its Applications[END_REF] and [START_REF] Ern | Finite Elements I: Interpolation and approximation[END_REF]), we regard afi n i t ee l e m e n ta sat r i p l e( K, P, Σ), where K is a non trivial polyhedron, P is a finite dimensional vector space of functions, typically polynomials K ! R or K ! R n , and Σ is a finite collection of linear functionals on P,c a l l e ddegrees of freedom.T h er o l eo fd e g r e e so ff r e e d o mi st w o f o l d :t h e ya l l o wt oi d e n t i f ya n d reconstruct elements of P by means of evaluations and also permit to glue local spaces defined on adjacent simplices.

In Section 1.1 we have introduced the concept of simplex and offered a compatibility structure on adjacent simplices that we have called simplicial complex. When the polyhedron K is a simplex T the corresponding structure is usually referred to as simplicial finite element [START_REF] Ern | Finite Elements I: Interpolation and approximation[END_REF]. In Section 1.3 we have constructed local spaces P r Λ k (T )oftrimmedpolynomialdifferen tialformsandofferedmoments as abasisfor P r Λ k (T )

⇤ ,th usdefiningmeaningfuldegreesoffreedom. Momentsfor an element of P r Λ k (T )in v olv ein tegralson`-subsimplices of dimension k  ` n paired with other appropriate local spaces of polynomial differential forms. This chapter is devoted to the construction and the study of alternative degrees of freedom for P r Λ k (T )c a l l e dweights,w h i c he x t e n dt h ec o n c e p to fL a g r a n g i a n finite elements from the nodal to the simplicial case. A weight is simply the integral of a differential k-form over a domain of dimension k.I np a r t i c u l a r ,w h e n k = 0 one retrieves nodal evaluations. When r =1 ,o n ej u s tt a k e si n t e g r a l so v e r k-subsimplices of T to obtain a valid set of linear functionals on P 1 Λ k (T ). When r>1oneneedsto(virtually)splitT into finer parts called small simplices.I ft h i s refinement is sufficiently regular and fine, it is possible to prove that the associate weights give a system of generators, but not a basis, for P r Λ k (T ) ⇤ .I nt h i sc a s e we shall talk about unisolvent but not minimal weights. The main result of this chapter consists in identifying a unisolvent and minimal subset of weights inherited from small simplices. This construction is intimately related with the selection of ab a s i sf o rP r Λ k (T ).

As o l i dm o v i n gr e a s o nf o rd e a l i n gw i t hw e i g h t si n s t e a do fm o m e n t si st h ef o llowing: suppose to be given a measurement of a physical quantity which is read by an instrument as a result of an experiment. For instance, if the entity under consideration is the electric field E,w h i c hi st h ep r o x yo fa1 -f o r m( s e e[ 1 7 ] ) ,o n e may read its circulation along a curve c as R c E • t c .T h ei n t e r p o l a t i o no ft h efi e l d E in P r Λ 1 (T )i se v i d e n ti nt h ec o n t e x to fe d g ee l e m e n t s [ 5 9 ] .

This chapter contains crucial results for the whole thesis. In particular, we here identify a unisolvent and minimal subset of small simplices. We first recall usual definitions of small simplices [START_REF] Rapetti | Whitney forms of higher degree[END_REF] and prove their main features [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF]. The associated weights are unisolvent but not minimal. Since small simplices offer some symmetry, there is not an evident geometrical reason for eliminating a specific collection rather than another. We hence define a different family of simplices allowing overlappings. This set still gives unisolvent weights and simplices of this family accumulate on specific faces of T .W et h u sr e m o v es i m p l i c e st h a tb r e a kt h e symmetry and obtain a unisolvent collection of weights. A dimension count shows that this family is now minimal. The parametrisation that describes this set is the same that distinguishes a basis from a system of generators for P r Λ k (T ). We thus remove the corresponding elements from the set of non overlapping small simplices and show that the resulting family is unisolvent and minimal.

The structure of Chapter 2 is as follows. It is subdivided into seven main sections. In the first we introduce basic concepts of weights and unisolvence. Each section from 2.3 to 2.6 aims to prove unisolvence of a different set of simplices; here a special attention is given to specific and extreme cases, usually associated with k =0 ,k = n and r =1 . T h el a s to n ei sd e v o t e dt os o m ea p p l i c a t i o n sa n d special cases. It is worth noting that there is a drop-down effect: Section 2.3 offers a tool needed for proving the main result of Section 2.4. Such a proposition is then needed to conclude Section 2.5, which is then used for proving the main result of the chapter, which is contained in Section 2.6.

We b egin by providing a formal and general definition of the claimed degrees of freedom that we call weights and we address the main feature we want to impose on them, which are unisolvence and minimality. This is Section 2.1, which is closed by restating some facts seen in Chapter 1 in this language.

We then apply the concept of weight looking at some more specific sets. In Section 2.2 we introduce the well known set of small simplices,r e i n t e r p r e t i n g the definition of [START_REF] Rapetti | Whitney forms of higher degree[END_REF]. This set is denoted by X k r (T ). We offer and formalise a description of elements of X k r (T )intermsofpairswhichmakesiteasytoenumerate them. We relate this definition with the classical one, which involves the map ξ that we introduce in (2.2). This equivalent construction puts us in a position to Chapter 2

show unisolvence of weights associated with X k r (T ). This is Proposition 2.17 and is the main result of Section 2.3. This result is not sharp, since #X k r (T ) P r Λ k (T ), as pointed out in (2.7). Cases where equality holds are discussed.

The following sections are then devoted to sharpening Proposition 2.17. In particular, Section 2.4 contains the formal construction of a set which has the same number of elements of X k r (T ) but allows overlappings. It hinges on the generalisation of the map ξ to another map, denoted by ⌧ ξ .W e c a l l t h i s s e t Σ k r (T ). We extend preliminary results of Section 2.3 and prove unisolvence of Σ k r (T )i nP r o p o s i t i o n2 . 2 0w i t has i m i l a rt e c h n i q u e . F r o mt h ep r o o f ,o n es e e s that the role of overlappings in Σ k r (T )i st om a k ec l e a rw h i c ho fi t se l e m e n t sa r e redundant in terms of unisolvence. Following the multi-index notation adopted for constructing a basis of P r Λ k (T )i nC h a p t e r1 ,E q u a t i o n( 1 . 2 5 ) ,w eo b s e r v et h a t the redundant elements of Σ k r (T )c o i n c i d ew i t ht h ed i s c a r d e dm u l t i -i n d i c e s . T h i s opens the way to the next section.

Section 2.5 contains the first complete construction of a unisolvent and minimal set, which is Σ k r,min (T ). This set has a strong theoretical relevance although its numerical performance, which is studied in Chapter 4, disqualifies it for applications. Its formal definition is provided in (2.16). Minimality, which is stressed in (2.17), is achieved by construction indexing elements of Σ k r,min (T )a sab a s i so fP r Λ k (T ). The strength of this set consists in allowing overlappings. This drives the choice of simplices and the relative unisolvence is Proposition 2.25. Techniques adopted now move away and exploit those needed for dealing with sets X k r (T )a n dΣ k r (T ). Thanks to the results of Section 2.5, exploiting the same enumeration of simplices that we use for Σ k r,min (T ), we identify a minimal set X k r,min (T ) ⇢ X k r (T ), see Equation (2.18). This construction ensures minimality since #X k r,min (T )= #Σ k r,min (T ). This is the content of Section 2.6, in which we prove one of the main results of this work, namely the unisolvence of X k r,min (T ), stated in Theorem 2.28. We conclude the chapter with Section 2.7 by offering two other sp ecial cases: k =1a n dk = n 1. In particular, the first one, which heavily hinges on the linearity of integral, allows to develop different and more direct proof for the case k =1 . I ti sw o r t hn o t i n gt h a ti fo n ec o n fi n et h e m s e l v e st ot h et h r e ed i m e n s i o n a l case, then these considerations put them in a position to prove directly all needed cases of unisolvence.

Weights

Definition 2.1 (Weight). Let s be a k-simplex supported in T and let ! 2 Λ k (T ). The weight of ! over s is Z s !. (2.1)
Considering ! 2P r Λ k (T ), since s is supported in T ,i tf o l l o w sf r o mt h el i nearity of the integral that the mapping

! 7 ! Z s ! is indeed an element of P r Λ k (T )
⇤ ,t h ed u a ls p a c eo fP r Λ k (T ). Weights as degrees of freedom for the spaces of high order Whitney forms were introduced first in [START_REF] Rapetti | Whitney forms of higher degree[END_REF], although some preliminary computations for the edge case (that is, for the case k = 1) already appeared in [START_REF]High order edge elements on simplicial meshes[END_REF]. The motivation for weights comes from physics and was first pointed out by Bossavit in his studies on computational electromagnetism [START_REF]Computational electromagnetism[END_REF]. In fact, in contrast with other degrees of freedom, weights offer an immediate physical interpretation as circulation along edges, fluxes across surfaces and densities in volumes. Moreover, since they involve only chains whose dimension equals the order of the form, they reduce to classical nodal evaluation considered in interpolation for k =0.

With any k-simplex s we thus associate a weight and, as degrees of freedom, we ask them to bear some features. The first requirement is unisolvence.

Definition 2.2 (Unisolvence). Let S k
r be a collection of k-simplices contained in T . The family of weights associated with S k r is said to be unisolvent for

P r Λ k (T ) if, for each ! 2P r Λ k (T ), one has that Z s ! =0 8s 2 S k r implies that ! =0.
To stress the supp ort of a family of k-simplices we will usually use notations like S k r (T ). Notice that with any family of k-simplices is naturally associated the corresponding family of weights. For the sake of brevity, when there is no ambiguity, with a slight abuse of notation we shall frequently say that a family of k-simplices is unisolvent to mean that the collection of weights associated with that family of k-simplices is unisolvent.

Unisolvence imposes a lower bound on the cardinality of the set S k r (T ), which shall be at least dim P r Λ k (T ). On the other hand, adding another simplex to a unisolvent set clearly does not affect its unisolvence. For this reason we shall look for minimal unisolvent sets, so that the associated set of weights is a basis for

P r Λ k (T ) ⇤ . Definition 2.3 (Minimality). The unisolvent set S k r (T ) for P r Λ k (T ) is said to be minimal if #S k r (T ) = dim P r Λ k (T ).
We now recall and extend known results about unisolvence and minimality. When r =1t h es p a c eP r Λ k (T ) reduces to usual Whitney forms [START_REF] Whitney | Geometric Integration Theory[END_REF] and the most natural unisolvent set for P 1 Λ k (T )i st h a to fk-subsimplices of T ,w h i c hw e denote by ∆ k (T ). This was proved in Proposition 1.22 which may be rewritten in the language of weights as follows.

Lemma 2.4. Integration on ∆ k (T ) gives unisolvent and minimal weights for

P 1 Λ k (T ).
For an alternative and complete pro of of this fact, which is not immediate, we address the reader to [START_REF]High order edge elements on simplicial meshes[END_REF]. Since there is a natural bijection between k-subsimplices of an n-simplex T and Whitney forms, this set is also minimal.

Small simplices

X k r (T ) Figure 2.1: The set X k r (T ), for r =3.
When r>1o n en e e d st oi n t r o d u c eal a r g e rn u m b e ro fk-simplices supported in T .T h efi r s ta t t e m p tc o n s i s t si ni n t r o d u c i n gsmall simplices,l o o s e l ys p e a k i n g afi n e rs u bd i v i s i o no fT into homothetic n-simplices (and relative k-subsimplices), see [START_REF] Rapetti | Whitney forms of higher degree[END_REF]. Unisolvence of particular families of small simplices has been studied in [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF]. However, as we shall make clear later, the problem of minimimality persists.

We offer an answer, identifying a subset of small simplices which is minimal for each k.

Let T =[ x 0 ,...,x n ]b ea nn-simplex. Recall that any x 2 R n can be represented by means of barycentric coordinates as

x = n X i=0 i x i
and such a representation is unique if we impose that P n i=0 i =1 . P o i n t sx 2 T are such that i (x) 0f o re a c hi =0 ,...,n.T h e f o l l o w i n g d e fi n i t i o n w a s introduced by Nicolaides in [START_REF] Nicolaides | On a class of finite elements generated by Lagrange interpolation[END_REF]. Definition 2.5 (Principal lattice). Let r 1. The principal lattice of order r of T is the set of points

L r (T ) . = ⇢ x 2 T : i (x) 2 ⇢ 0, 1 r ,..., r 1 r , 1 for i =0,...,n .
A useful subset of the principal lattice, which we will extensively use in our proofs, is

Z r (T ) . = {x 2 L r (T ): 0 (x) 6 =0} .
Observe that Z r (T )=L r 1 (T 0 ), being T 0 =[y 0 ,...,y n ]ann-simplex which is r 1 r homothetic to T and such that y 0 coincides with x 0 . points. Of course these sets are meaningful when T is replaced by one of its k-faces F . We are in a p osition to define the set of small k-simplices, revisiting the standard definition of [START_REF] Rapetti | Whitney forms of higher degree[END_REF]. For each ξ 2 R n consider the map ξ which is the composition of a homothety of factor 1 r that reads as x 7 ! x 0 + 1 r (x x 0 )a n dt h e translation given by x 7 ! x +(ξ x 0 ), which is readily computed as

x 1 x 2 x 0 ⌘ y 0 y 1 y 2 • • • •
ξ : x 7 ! ξ (x)= 1 r (x x 0 )+ξ. (2.2)
Definition 2.6 (Small simplex). Let T be an n-simplex and let ∆ k (T ) denote the collection of k-subsimplices of T , with 0  k  n. The set of small k-simplices of order r of T is

X k r (T ) . = { ξ (F ):F 2 ∆ k (T ), ξ 2 Z r (T )} .

Chapter 2

Likewise, for any k 0 -face F of T , we put

X k r (F ) . = { ξ (G):G 2 ∆ k (F ), ξ 2 Z r (F )} , provided that 0  k  k 0 .
The case k = 0 requires some additional attention, since two elements of X k r (T ), for k>0, may intersect in a node of X 0 r (T ). For this reason we shall frequently study and treat it separately. Also, it will be motivating for several examples and choices that we will make.

We restate Definition 2.6 in terms of a pair (F, α) 2 ∆ k (T ) ⇥I(n +1,r 1). In [START_REF]High order edge elements on simplicial meshes[END_REF] this construction has been given for small edges (i.e. for k =1)andhasbeen generalised in [START_REF] Rapetti | Whitney forms of higher degree[END_REF] to any k 2 N.I nb o t h[ 5 9 ,S e c t i o n3 .1] and [60, Section 3.1] it is introduced componentwise, and it is given the name of e k map.W ep r e s e n ti t here as a map that sends a k-subsimplex F 2 ∆ k (T )o n t oak-simplex supported in T .F o rt h i sr e a s o ni ti sc o n v e n i e n tt or e a di ti nb a r y c e n t r i cc o o r d i n a t e s .F o r each α 2I (n +1,r 1) we define a mapping that transforms the barycentric coordinates of x 2 T as

:( 0 (x),..., n (x)) 7 ! 1 r ( 0 (x)+↵ 0 ,..., n (x)+↵ n ) . (2.3)
Then small k-simplices are (see [START_REF]High order edge elements on simplicial meshes[END_REF]

, Definition 3.2]) pairs s =( F, α)w i t h F 2 ∆ k (T )a n dα 2I(n +1,r 1).
Example 2.7. Consider the standard 2-simplex T ⇢ R 2 . We have I(3, 1) = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} .

Any element of ∆ 1 (T ) is parametrised by a permutation : {0, 1}!{ 0, 1, 2}.

Denote by E i the edge opposite to the i-th vertex of T , so that E 0 =[ e 1 , e 2 ], and so on. The elements of X 1 2 (T ) generated by E 0 are {(E 0 , (1, 0, 0)) , (E 0 , (0, 1, 0)) , (E 0 , (0, 0, 1))} , those generated by E 1 are

{(E 1 , (1, 0, 0)) , (E 1 , (0, 1, 0)) , (E 1 , 0, 0, 1))} and those generated by E 2 are {(E 2 , (1, 0, 0)) , (E 2 , (0, 1, 0)) , (E 2 , (0, 0, 1))} , see Figure 2.
3 for a depiction. We show that, for instance, (E 0 , (1, 0, 0)) represents a segment parallel to E 0 . In barycentric coordinates, we have (e 1 )=(0, 1, 0) and (e 2 )=( 0 , 0, 1). Thus (2.3) yields (e 1 )= 1 2 (1, 1, 0) and (e 2 )= 1 2 (1, 0, 1), hence the small 1-simplex associated with (E 0 , (1, 0, 0)) has vertices of barycentric coordinates (with respect to T ) (e 1 ) and (e 2 ). In Cartesian coordinates they correspond to 1 2 , 0 and 0, 1 2 , whence the claim.

Writing (2.3) in Cartesian co ordinates, we read any s 2 X k r (T )i nt e r m so ft h e vertices {x 0 ,...,x n } of T .I nf a c t ,g i v e nF 2 ∆ k (T )a n dα 2I(n +1,r 1), we have

s = ⇢ p 2 T : p = 1 r x +[x 0 | ...|x n ] α T , x 2 F , (2.4)
where the quantity [x 0 | ...|x n ] α T is to be intended as a vector v =

P n i=0 ↵ i x i 2 R n .
Remark 2.8. Equation (2.4) outlines the relationship between such multi-index α and the vector ξ in Definition 2.6 thought of as element of Z r (T ). In fact, if one matches (2.4) with Equation (2.2), one gets the relationship

ξ = 1 r x 0 +[x 0 | ...|x n ] α T
All these facts provide an explicit description of elements of X k r (T )inte rmsof a k-subsimplex F of T and a multi-index α.T h i sd e s c r i p t i o ni sc o m p a t i b l ew i t h the boundary operator @.T h i ss i m p l ef a c tw i l lb ee x t e n s i v e l yu s e di nf o r t h c o m i n g proofs, using elements of

X k 1 r (T )toconstructtheboundaryofelementsofX k r (T ). Lemma 2.9. If s 2 X k r (T ) then @s is a formal linear combination of elements of X k 1 r (T ).
Proof. This is basically due to the linearity of the boundary operator and the fact that the boundary @s of any s =[x (0) ,...,x (k) ] 2 ∆ k (T )m a ybew r i t t e na st h e formal combination of elements of ∆ k 1 (T ).

Consider s =(F, α)w i t hF 2 ∆ k (T )a n dα 2I(n +1,r 1). Then

s = 1 r F + 1 r [x 0 | ...|x n ] α T whence @s = 1 r @F + 1 r [x 0 | ...|x n ] α T =(@F, α)= [ E2@F (E, α).
The union is disjoint for k>0, see next example.

We give an explicit idea of the statement of Lemma 2.9. Consider X 2 2 (T ), being T a2 -s i m p l e xo fe d g e sE `=[ x i , x j ], with distinct i, j, `=1 , 2, 3. The set X 2 2 (T ) consists of three 2-simplices, associated with pairs (T,(1, 0, 0)), (T,(0, 1, 0)) and (T,(0, 0, 1)). Boundaries shall be formed by elements of X 1 2 (T ), that are those computed in Example 2.7. Their placement is depicted in Figure 2.3.

Representations of vertices of small simplices in X k r (T ) in terms of pairs (F, α) are not unique, as Example 2.10 shows. Loosely speaking, we show that it is possible to obtain more than one representation for the same element of X 0 r (T ). The nodal case, however, motivates the subsequent construction.

x 0 x 1 x 2 (T, (1, 0, 0)) (T, (0, 1, 0)) (T, (0, 0, 1)) (E 2 , (1, 0, 0)) ( E 0 , ( 1 , 0 , 0 ) ) ( E 1 , ( 1 , 0 , 0 ) ) (E 2 , (0, 1, 0)) ( E 0 , ( 0 , 1 , 0 ) ) ( E 1 , ( 0 , 1 , 0 ) ) (E 2 , (0, 0, 1)) ( E 0 , ( 0 , 0 , 1 ) ) ( E 1 , ( 0 , 0 , 1 ) ) Figure 2.3: Elements of X 1 2 (T )formtheboundaryofelementsofX 2 2 (T )andrespect the associated indices. Example 2.10. Consider the 2-simplex T =[ x 0 , x 1 , x 2 ] ⇢ R 2 . Midpoints of edges [x i , x j ] belong to X 0 1 (T )
, since they satisfy Definition 2.6 for r =1and k =0 . However they are associated with more than a pair (x `, α). In fact, take for instance m as the midpoint of the edge [x 0 , x 1 ]. One has that m =(x 0 , (0, 1, 0)) but also m =(x 1 , (1, 0, 0)). This is depicted in Figure 2.4. Recall that any F 2 ∆ k (T )i sg e n e r a t e db yk +1 v e r t i c e s o f T ,i . e . F = [x (0) ,...,x (k) ], hence any F 2 ∆ k (T ) is associated with an increasing permutation : k ! n. It follows from above example that one gets repetitions of 0-simplices whenever ↵ i 6 =0f o rs o m ei< (0). Now, recall the set defined in (1.25):

x 0 x 1 x 2 m (x 0 , (0, 1, 0)) (x 1 , (1, 0, 0))
I (n +1,r 1) . = {α 2I(n +1,r 1) : ↵ i =0for i< (0)} .
In view of Example 2.10, in order to avoid repetitions of 0-simplices in Definition 2.6 for the case k =0 ,w ed ot h ef o l l o w i n g . W eo b s e r v et h a ta n yp o i n t is indexed by an integer j such that (0) = j,t h u sw em a yi d e n t i f y with its image j and write I j (n +1,r 1) in place of I (n +1,r 1). Then, instead of considering the totality of elements of I(n +1,r 1), for each x j we may let α vary in the set I j (n +1,r 1), being in fact j = (0). Collecting all these pairs (x j , I j (n +1,r 1)) in the set e I(n +1,r 1) we still get the set X 0 r (T )=L r (T ) and hence X 0 r (F )=L r (F ). The following dimension count shows that we are in fact not missing any node. It makes explicit the case k =0ofLemma1.18.

Lemma 2.11. Let be a permutation that represent a 0-simplex, i.e. : {0}!n. One has

# e I(n +1,r 1) = ✓ n + r r ◆ .
Proof. By direct computation, observing that the cardinality of I(n +1,r 1) is given by the sum of the cardinalities of the sets I(k +1,r 1) for k =0,...,n 1.

# e I(n +1,r 1) = n X i=0 #I i (n +1,r 1) = n X i=0 ✓ (n i)+(r 1) r 1 ◆ = n + r r ✓ n + r 1 n ◆ = n + r r (n + r 1)! (r 1)!n! = (n + r)! r!n! = ✓ n + r r ◆ .
The claim is proved.

We compute the cardinality of the set X k r (T )o fs m a l lk-simplices of T .

Lemma 2.12. For k>0 and r 1, one has

#X k r (T )= ✓ n +1 k +1 ◆✓ n + r 1 r 1 ◆ . (2.5)
For k =0, one has

#X 0 r (T )= ✓ n + r r ◆ . (2.6) Chapter 2
Proof. Assume k>0. Since, by construction, elements of X k r (T )in tersectatmost in vertices, one has that each element of X n r (T )c o n t r i b u t e st oX k r (T )w i t ha l li t s k-simplices. Hence #X k r (T )c o i n c i d e sw i t ht h en u m be ro fk-subsimplices of an nsimplex times the number of small n-simplices of order r,w h i c hi st h ec a r d i n a l i t y of Z r (T ). Thus

#X k r (T )= ✓ n +1 k +1 ◆✓ n + r 1 r 1 ◆ .
For k =0w es h a l lo b s e r v et h a tt h ea b o v ee q u a t i o nd o e sn o th o l da n ym o r e , since elements of X n r (T )i n t e r s e c ti np o i n t so fX 0 r (T ). The sets X 0 r (T )a n dL r (T ) coincide by construction (see Lemma 2.11 and the preceeding example), hence one has

#X 0 r (T )=#L r (T )= ✓ n + r r ◆ .
This concludes the proof.

Remark 2.13. Notice that when r =1the latter term of (2.5) is equal to 1, hence the set of small k-simplices X k 1 (T ) coincides with the collection of k-subsimplices of T , ∆ k (T ).

The distinction between the cases k =0a n dk>0h a sad e e p e rm e a n i n g . Indeed, when k =0,thespaceP r Λ 0 (T )c a nb et h o u g h to fa sap o l y n o m i a ls p a c e P r (T )w h o s ee l e m e n t sa r eg i v e nb yt h ep r o d u c to fad e g r e e1p o l y n o m i a la n d another polynomial of degree r 1. In this case in fact Whitney 0-forms play the role of a usual degree 1 polynomial, hence points and associated nodal evaluations offer well known degrees of freedom.

Unisolvence of X k r (T )

Small k-simplices are a natural generalisation of k-subsimplices, in a sense that we will make precise in Proposition 2.17. A direct computation shows that, for k>0,

#X k r (T )= ✓ n +1 k +1 ◆✓ n + r 1 r 1 ◆ ✓ n + r n k ◆✓ r + k 1 k ◆ =dimP r Λ k (T ), (2.7)
where equality is reached only when k = n or r =1 . S e v e r a lr e s u l t sa b o u tt h e set X k r (T ), in particular its unisolvence, were proved in [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF]. The main strategy in proving this kind of results, which we will frequently adopt throughout this work, consists in converting the problem of integrating over k-simplices into that of evaluating appropriate polynomials on points of lattices. Typically one then concludes applying a lemma or a collateral result which bridges the relationship between integrals and the polynomial. This simplifies things as one may then invoke results on interpolation of functions, which have been extensively studied by many authors.

The first aim of this section is to prove unisolvence of the set X k r (T )f o r P r Λ k (T ), which is here Proposition 2.17. We premise few technical results. Although they are known, we prove them as we shall generalise them with similar techniques in the sequel. Lemma 2.14 ([21], Lemma 3.12). Let ! 2P r Λ n (T ) be such that

Z T ⇤ ξ ! =0 8ξ 2 R n .
Then ! =0.

Proof. By the change of variable, we have that

0= Z T ⇤ ξ ! = Z ξ (T )
!.

Now, identify the n-form ! with its coefficient, which is a polynomial p(x). The zero locus of p(x)i sac l o s e ds e to fm e a s u r e0 ,h e n c et h e r ee x i s t ss o m eξ 2 R n for which ξ (T )d o e sn o ti n t e r s e c ts u c haz e r ol o c u s . I tf o l l o w st h a tt h es i g no fp(x) is constant in that region, hence p(x)=0,th us! =0.

Recall that Π k T :Λ k (T ) !P 1 Λ k (T )d e n o t e st h ei n t e r p o l a t i o no p e r a t o ro n t o lowest order Whitney forms, defined in (1.32), which associates to any ! 2 Λ k (T ) that Whitney form which has the same weight as ! on the k-subsimplices of T . Its features have been discussed at the end of Chapter 1.

Lemma 2.15 ([21], Lemma 3.13). Let T ⇢ R n be an n-simplex. Let ! 2P r Λ k (R n ), with 0 <k<n.I fΠ k T ⇤ ξ ! =0for each ξ 2 R n , then ! =0.
Proof. Suppose F 2 ∆ k (T )i sak-face of T .T h e nb yh y p o t h e s i s

Z F Π k T ⇤ ξ ! = Z F ⇤ ξ ! =0 8ξ 2 R n
that are tangent to F . Now, Lemma 2.14 applied to F and its tangent space implies that the pullback of ! to F is zero. Observe that all translates of F have the same tangent space (as a vector space), hence the pullback of ! to any affine k-dimensional subspace of R n which is parallel to F vanishes as well. Since this holds for all k-faces F 2 ∆ k (T ), we have that the pullback of ! vanishes on any affine subspace which is parallel to a k-face of T .T h i si m p l i e s , in particular, that Π k T 0 ! =0f o re a c hF 0 obtained from a k-face of T by rescaling and translation. Now, if x 2 R n ,t h e r ee x i s t sas i m p l e xT 0 =[y 0 ,...,y n ]w h i c hi s obtained from T by rescaling and translation and is such that x = y 0 . Now observe that, since x is a vertex of T 0 ,t h et a n g e n ts p a c et oT in x t x R n ⇠ = R n is spanned by t x F 0 ,w h e r eF 0 is a face of T 0 which contains x and is parallel by construction to a face of T . Writing each of ξ 1 ...,ξ k 2 t x R n as (non unique) sum of elements in t x F 0 ,b ym u l t i l i n e a r i t yo fd i ff e r e n t i a lf o r m sw e get ! =0.

Remark 2.16. We claimed the results of Lemma 2.14 and Lemma 2.15 for polynomial forms P r Λ k (T ). However, since we have no restrictions on the degree r and we have inclusions

P r Λ k (T ) ✓P r Λ k (T ),
such results naturally apply to P r Λ k (T ) as well.

Notice that, in the above proof, there is some freedom in the choice of T 0 ;i n particular, when we look at the faces F 0 involved in the last part of the proof, we do consider all k-faces of T .W ew i l ls h a r p e nt h i sr e s u l tl a t e ro n .

The above lemmas are needed to prove the following general result on unisolvence of small simplices. We follow its original proof, which is given in [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF].

Proposition 2.17 ([21], Proposition 3.14). The set of small simplices X k r (T ) is unisolvent for P r Λ k (T ).

Proof. We shall prove that, for each

! 2P r Λ k (T ), if Z s ! =0 8s 2 X k r (T )
it follows that ! =0. We carry out the proof by a descending induction. Assume k = n. Since ! 2 P r Λ n (T )i sc l o s e df o rd i m e n s i o n a lr e a s o n s ,i tf o l l o w st h a ti nf a c t! 2P r 1 Λ n (T ). Hence the map

R n 3 ξ 7 ! q(ξ)= Z T ⇤ ξ !
is a polynomial of degree r 1w h i c hv a n i s h e s ,b yh y p o t h e s i s ,o nt h ev e c t o r s ξ 2 Z r (T )( t h o u g h to fa sp o i n t si nR n ), which form a principal lattice of order r 1, as already observed. It follows that q(ξ)=0foreac hξ 2 R n ,h e n c eL e m m a 2.14 implies ! =0.

Assume now k<n .W ew a n tt os h o wt

h a ti f! 2P r Λ k (T )i ss u c ht h a t Z s ! =0 8s 2 X k r (T ), (2.8)
then ! =0 . S i n c e! 2P r Λ k (T )i t sd i ff e r e n t i a ld ! 2P r Λ k+1 (T ). Inductive hypothesis ensures that the result holds true for any k<k 0  n and, in particular, for k + 1. Thus, by Lemma 2.9, applying Stokes' Theorem we have that

X s2@S Z s ! = Z @S ! = Z S d! =0 8S 2 X k+1 r (T )
for an appropriate collection of elements of X k r (T ). This implies that d! =0 , whence ! 2P r 1 Λ k (T )b yP r o po s i t i o n1 . 1 1 ,(iii).

We show that any Combining Proposition 2.17 and equation (2.7) we showed that the spaces X k r (T ) are associated with unisolvent but not minimal weights for P r Λ k (T ), except when either k =0ork = n (for any r)o rr =1(foran yk). The first example of minimal and unisolvent set for such spaces has been characterised in [1], where it also appears without proof a rule for extracting a subset X k r (T )w i t ht h ed e s i r e d properties. We reconstruct that set and prove the claimed features.

! 2P r 1 Λ k (T )obeying(2.8)isthezeroform. Bydefinition, any s 2 X k r (T )i
We generalise the map ξ in (2.2), namely

ξ (x)= 1 r (x x 0 )+ξ,
by letting the factor1 r vary as

⌧ ξ : x 7 ! ⌧ ξ (x)= 0 (ξ)(x x 0 )+ξ. (2.9) Then, expanding ξ = P n i=0 i (ξ)x i , ⌧ ξ 1 is easily manipulated to ⌧ ξ (x)= 0 (ξ)(x x 0 )+ n X i=0 i (ξ)x i = = 0 (ξ)x + n X i=0 i (ξ)x i 0 (ξ)x 0 = 0 (ξ)x + n X i=1 i (ξ)x i .
When applied to T ,foran yξ 2 R n this map yields an n-simplex with the property that the face opposite to the image of x 0 belongs to the hyperplane containing

F 0 . =[ x 1 ,...,x n ].
We may then replace the map ξ by ⌧ ξ in Definition 2.6 and produce the following sets. We put

Σ k r (T ) . = {⌧ ξ (F ):F 2 ∆ k (T ), ξ 2 Z r (T )} (2.10)
and similarly

Σ k r (F ) . = {⌧ ξ (G):G 2 ∆ k (F ), ξ 2 Z r (F )} , (2.11)
provided that F is a k 0 -face of T ,w i t h0 k  k 0 . An example of a set of the type (2.10), compared with the corresponding X k r (T ), is shown in Figure 2.6.

x 0 x 1 x 2 x 0 x 1 x 2 Figure 2.6: On the left, the set X 2 2 (T )f o rT ⇢ R 2 is depicted.
On the right, the set Σ 2 2 (T ) in the same setting is represented. Notice that its element overlap on the lowest triangles (hence they are shaded more heavily).

We provide a characterisation of the sets Σ k r (T )w h i c hf o l l o w st h a to fX k r (T ). Observe that elements of the sets X k r (T )a n dΣ k r (T )a r eb o t ha s s o c i a t e dw i t ha k-face of T and an element of the principal lattice Z r (T ). It follows that there is a natural bijection between the elements of X k r (T )a n dt h o s eo fΣ k r (T ), as well as one between X k r (F )a n dΣ k r (F ). Lemma 2.14 and Lemma 2.15 then generalise as follows, and note again that we prove Lemma 2.18 for the space P r Λ k (T ) and that its validity for high order Whitney forms is a consequence of Remark 2.16.

Lemma 2.18. Let ! 2P r Λ n (T ) be such that Z T ⌧ ⇤ ξ ! =0 8ξ 2 R n .
Then ! =0.

The proof of this fact is close to that of Lemma 2.14, although it requires a bit more caution on some technical details.

Proof. Since ! is an n-form, we may identify it with its coefficient, say the polynomial q.B yh y p o t h e s i s ,w eh a v e 0=

Z T ⌧ ⇤ ξ ! = Z T q 0 (ξ)x + n X i=1 i (ξ)x i ! dx. (2.12) Let h F 0 . = {ξ 2 R n | 0 (ξ)=0}
be the affine hyperplane containing F 0 .D e n o t e by q 1 (0) the zero locus of q.W efi r s ts h o wt h a th F 0 ⇢ q 1 (0), i.e. ! vanishes on h F 0 .

Observe that for any ξ 2 h F 0 ,E q u a t i o n( 2 . 1 2 )r e a d sa s 0=

Z T q n X i=1 i (ξ)x i ! dx.
In particular q h F 0 is a constant polynomial with respect to x,t h u st h ea b o v e equation implies that q h F 0 =0,henceh F 0 ⇢ q 1 (0). Pick now ξ such that 0 (ξ) 6 =0. Then⌧ ξ induces a change of variable

y = 0 (ξ)x + n X i=1 i (ξ)x i , (2.13
)

whence dy = | 0 (ξ)| n dx.P u tt h e nT ξ . = ⌧ ξ (T )=[y 0 ,...,y n ]. Thus 0= Z T q 0 (ξ)x + n X i=1 i (ξ)x i ! dx = | 0 (ξ)| n Z T ξ
q(y)dy, hence (2.12) vanishes if and only if R T ξ q(y)dy does. We show that there exists a full measure region of R n that contains T ξ in which the sign of q is constant. It will follow that ! ⌘ 0o nR n .

In order to compute such an integral, we study T ξ ,s ow ee x a m i n et h ea c t i o n of (2.13) on vertices of T .F o re a c hξ 2 R n ,w eh a v e

y j = 0 (ξ)x j + n X i=1 i (ξ)x i = 1 n X i=1 i (ξ) ! x j + n X i=1 i (ξ)x i = x j + n X i=1 i (ξ)(x i x j ).
As a consequence a vertex y j of T ξ lies on h F 0 if and only if j 6 =0. Itfollo wsthat any T ξ is a scaling of T by factor 0 (ξ) translated along h F 0 .

Since h F 0 is a hyperplane, it is described by a polynomial of degree one p. Since h F 0 ⇢ q 1 (0), there exists m 2 N, m>1s u c ht h a tq = p m q;w ec h o o s es u c hm to be maximal, i.e. such that p does not divide q.

We claim that there exist x 2 h F 0 and r>0s u c ht h a tB(x,r)

\ q 1 (0) = B(x,r) \ h F 0 ;w es h o wt h ed o u b l ei n c l u s i o n . O n ei st r i v i a l ,a sh F 0 ⇢ q 1 (0) yields B(x,r) \ h F 0 ⇢ B(x,r) \ q 1 (0).
To prove the other inclusion we work by contradiction. We then assume that

B(x,r) \ h F 0 ⇢ B(x,r) \ q 1 (0) 8x 2 h F 0 , 8r>0,
i.e. that there exists y 2 (q 1 (0) \ h F 0 )f o re a c hx 2 h F 0 and r>0. Since we are assuming that y 6 2 h F 0 ,itm ustbey 2 q 1 (0). For each x 2 h F 0 we may construct asequenceofpoin ts{y `}`2N ⇢ q 1 (0) such that y `! x. Since q 1 (0) is closed, by the sequential closure Theorem, it follows that x 2 q 1 (0) for each x 2 h F 0 .T h u s h F 0 ⇢ q 1 (0), which contradicts the assumption of maximality on m,theexponen t of p.

As stressed in Remark 2.16, Lemma 2.18 applies to P r Λ k (T )a sw e l l . T h e following Lemma is a generalisation of Lemma 2.15 and for 0 (ξ)= 1 r one in facts retrieves that result.

Lemma 2.19. Let T ⇢ R n and let ! 2P r Λ k (T ), with 0 <k<n , be such that

Π k T (⌧ ⇤ ξ !)=0for all ξ 2 R n . Then ! =0.
Proof. The proof of this fact is identical to that of Lemma 2.15, except in the fact that we consider the k-form

⌧ ⇤ ξ ! = !( 0 (ξ)x + P n i=1 i (ξ)x i )
. Hence, we choose as basis for T x R n only vectors that do not lie on F 0 .T h er e s u l tt h e nf o l l o w s .

We are ready to generalise Proposition 2.17 to this context, proving that integration over elements of Σ k r (T )g i v e su n i s o l v e n tw e i g h t sf o rP r Λ k (T ).

Proposition 2.20.

Let ! 2P r Λ k (T ).I f Z s ! =0 8s 2 Σ k r (T ), then ! =0.
Proof. We retrace the pro of of Prop osition 2.17 and pro ceed by descending induction. Suppose k = n.T h e n! is closed for dimensional reasons, thus in fact ! 2P r 1 Λ n (T ). The map that associates

R n 3 ξ 7 ! q(ξ)= Z T ⌧ ⇤ ξ !
is a polynomial of degree r 1w h i c hv a n i s h e s ,b yh y p o t h e s i s ,o nt h ep r i n c i p a l lattice that defines Σ r n (T ). Hence q(ξ)=0f o re a c hξ 2 R n . By Lemma 2.18 we get ! =0andthebasecaseofinductionispro v ed.

Assume now k<nand that the result holds for each k<k 0  n.S i n c et h e boundary of any element of Σ k+1 r (T )i st h ef o r m a ls u mo fe l e m e n t so fΣ k r (T ), we may apply Stokes' Theorem and we get

X s2@S Z ! = Z @S ! = Z S d! =0,
hence the inductive hypothesis yields that d! =0 ,h e n c e! is closed, so in particular ! 2P r 1 Λ k (T ). Recalling the definition of the projector Π k T ,w eh a v ea polynomial map ξ 7 ! q(ξ)=Π k T (⌧ ⇤ ξ !) which is of degree r 1inξ,since! is. Moreover, it vanishes by hypothesis on the principal lattice Z r (T ), since 0

= R T ⌧ ⇤ ξ ! = R T ⌧ ⇤ ξ (Π k T (!)) implies that Π k T (⌧ ⇤ ξ !)=0, since Π k T (⌧ ⇤ ξ !) 2P 1 Λ k (T ).
It follows that q is the zero polynomial, i.e. q(ξ)=0 for all (ξ) 2 R n .L e m m a2 . 1 9i m p l i e st h a t! =0,hencetheresultispro v ed.

Unisolvence and minimality of Σ k r,min (T )

In this and the following section we identify minimal and unisolvent sets inside the families X k r (T )a n dΣ k r (T )t h a tw eh a v ec o n s t r u c t e da tt h eb e g i n n i n go ft h e chapter.

To b egin with, we construct the minimal subset Σ k r,min (T )o fΣ k r (T ), since it will be needed to construct X k r,min (T ). To fix a concrete idea of this set, which motivates our choice under a geometrical point of view, consider the following example in T ⇢ R 3 .W es t u d yt h ed i s t r i b u t i o no fe l e m e n t so fΣ 2 r (T )inT ,assuming for simplicity r =3,anddepictthefaceF 0 ⇢ T in Figure 2.8.

We immediately note that the face F 0 contains 5 2 =10simplicesofdimension 2, despite we know from Proposition 2.20 that only 4 2 =6ofthemarenecessary for obtaining unisolvence. In particular, that result also tells us which 2-simplices lying on F 0 are unnecessary. Remark 2.21. Recall that F 0 =[x 1 ,...,x n ]. Hence, by construction, if F ✓ F 0 , any element of Σ k r (F ) is supported in F 0 . We associate each s 2 Σ k r (T )w i t hap a i r( F, α) 2 ∆ k (T ) ⇥I(n +1,r 1), where I(n +1,r 1) denotes the set of multi-indices in n +1 variables and weight |α| = r 1. Any s 2 Σ k r (T ) may be written in terms of the map (2.9) as

s = 8 > < > : p 2 T : p = ↵ 0 +1 r 0 B @ T +[x 1 | ...|x n ] 0 B @ ↵ 1 . . . ↵ n 1 C A 1 C A , x 2 F 9 > = > ; . (2.14)
This offers a definition of Σ k r (T )i nt e r m so ft w oe n t i t i e s ,t h ek-faces of T and the indices I(n+1,r 1), that are easy to handle and enumerate, hence it is immediate to deduce that #X k r (T )=#Σ k r (T )since,inbothcases,F and α range in the same sets. Now that we have this characterisation, we impose a condition on elements of Σ k r (T )b yw o r k i n go n∆ k (T )a n dI(n +1,r 1). Once any s 2 Σ k r (T )i si d e n t i fi e d with a pair (F, α) 2 ∆ k (T ) ⇥I(n +1,r 1) we define a subset Σ k r,min (T ) ✓ Σ k r (T ) such that where the subscript s (F,α) is introduced to stress the dependence of s on the above specified sets and F =[ (0),..., (k)]. For each of such increasing permutations, we put

s (F,α) 2 Σ k r,min (T ) () ↵ i =0foralli such that 1  i  (0), (2.15)
x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3 x 1
J (n +1,r 1) . = {α 2I(n +1,r 1) : ↵ i =0, 1  i  (0)} .
Once the increasing permutation is fixed, the cardinality of these sets coincides with that of I (n +1,r 1) defined in (1.25), which was computed in Lemma 1.18. This is immediate by shifting all indices in (1.25) by 1. We thus identify a k-face F 2 ∆ k (T )w i t hi t sd e fi n i n g( i n c r e a s i n g )p e r m u t a t i o n and consider the collection of pairs e J (n +1,r 1) . = [

F 2∆ k (T ) {(F, α) 2 ∆ k (T ) ⇥J (n +1,r 1)} ,
whose cardinality is again that computed in (1.27) in Lemma 1.18. We thus define

Σ k r,min (T ) . = n s (F,α) 2 Σ k r (T ):( F, α) 2 e J (n +1,r 1) o (2.16)
and, by construction, we have that #Σ k r,min (T )=dimP r Λ k (T ).

(2.17)

By construction we obtain the following Lemma.

Lemma 2.22. One has Σ n r,min (T )=Σ n r (T ) and Σ 0 r,min (T )=Σ 0 r (T )=L r (T ). Figure 2.9 shows the difference between Σ k r (T )andΣ k r,min (T )fork =1andr = 3i na2 -s i m p l e xT .I np a r t i c u l a r ,i te n l i g h t e n st h a tt h er e d u n d a n c ya c c u m u l a t e s on F 0 . The next proposition, together with (2.17), shows that Σ k r,min (T )isaunisolvent and minimal set for P r Λ k (T )f o re a c h0 k  n and r>0. The idea behind the proof consists in showing that the removal of the redundant simplices yields sets for which former results still apply. To this end some technical preliminaries are needed, as well as an extensive use of the set of indices as in (2.15). We thus stress those dependences, placing also a subscript in front of the k-subsimplices of ∆ k (T ) when needed. A proof of this fact first appeared in [1].

x 1 x 2 x 0 x 1 x 2 x 0
Lemma 2.23. Let F 2 ∆ k (T ) be the k-subsimplex of T associated with the per- mutation { (0),..., (k)}. The set Σ k r (F ) is thus well defined. If ! 2P r Λ k (F ) is such that Z s ! =0 8s 2 Σ k r (F ), then ! =0.
Proof. It follows immediately from Proposition 2.20 applied on F instead of T .

Lemma 2.24.

Let ! 2P r Λ k (T ) be such that Z s ! =0 8s 2 Σ k r,min (T ).
Then ! Fσ =0for each F 2 ∆ k (T ).

Proof. It suffices to show that Σ k r (F ) ⇢ Σ k r,min (T )f o ra l lF 2 ∆ k (T ). Notice that any element of Σ k r (F )isdescribedb yam ulti-indexθ 2I(k +1,r 1), since dim F = k.T h e n ,r e c a l l i n gt h a ta n ys 2 Σ k r,min (T )i sd e s c r i be db yap a i r( F , α), for each F 2 ∆ k (T )w ec o n s i d e rt h em u l t i -i n d e xα 2I(n +1,r 1)

↵ j = 8 > < > : ✓ 0 if j =0 ✓ i if j = (i), 1  i  k 0o t h e r w i s e . Then Σ k r (F ) 3 s (Fσ,θ) = s (Fσ,α) 2 Σ k r,min (T )b e c a u s e↵ i =0f o ra l li such that 1  i  (0). It follows that Σ k r (F ) ⇢ Σ k r,min (
T ). We are hence in a p osition to prove Prop osition 2.25. Assume then 0 <k<n.W es h o wt h a t

Proposition 2.25. Let ! 2P r Λ k (T ) be such that Z s ! =0 8s 2 Σ k r,min (
Z s ! =0 8s 2 Σ k r,min (T )= ) Z s ! =0 8s 2 Σ k r (T ),
hence the result will follow from Proposition 2.20. First we notice that

Σ k r (T ) \ Σ k r,min (T ) ⇢{F 2 ∆ k (T ): (0) > 0} . In fact, s (Fσ,α) 2 Σ k r (T )\Σ k r,min (
T ) if and only if there exists i such that 1  i  (0) and ↵ i 6 =0. I f (0) = 0 such an i does not exist. Hence F ⇢ F 0 and by Remark 2.21 we get s (Fσ,α) ⇢ F 0 .

Applying Lemma 2.24, we get that ! Fσ =0 ,h e n c ei np a r t i c u l a r! s (Fσ ,α) =0 for all s (Fσ,α) 2 Σ k r (T ) \ Σ k r,min (T ), thus

R s ! =0foralls 2 Σ k r (T )
. By Proposition 2.20 we have that ! =0.

It will be shown that the sets Σ k r,min (T )d on o to ff e re n c o u r a g i n gr e s u l t sf o r numerical purposes (see Table 4.13 on page 103, in the section dedicated to non simplicial supports). However, they are the key tool to derive several unisolvent and minimal sets of simplices, whence their theoretical relevance.

2.6 Unisolvence and minimality of X k r,min (T )

Figure 2.10: Left: the set X 2 r,min (T ), for r =3. Righ t: theredundan tsimplicesin X 2 r (T ), for r =3,i.e. thecollectionX 2 r (T ) \ X 2 r,min (T ).

We now turn to the main result of the chapter. We exploit the collection of the above lemmas and propositions to identify a minimal subset X k r,min (T ) ⇢ X k r (T ) which keeps the property of unisolvence.

To b egin with, we construct such a set X k r,min (T ). The set of indices invoked is the same used for identifying a basis of P r Λ k (T )a n dt h em i n i m a ls e tΣ k r,min (T ). This choice also extends that provided for avoiding double counting in the definition of X 0 r (T ). Notice that, in fact, the set X 0 r (T )i sn o to n l yu n i s o l v e n tf o r P r Λ 0 (T )=P r (T ) but also minimal, see Lemma 2.12.

Recall that any s 2 X k r (T )i sd e s c r i b e db yap a i r( F, α)w i t hF 2 ∆ k (T )a n d α 2I(n +1,r 1). For any F 2 ∆ k (T )d e fi n e db yt h ep e r m u t a t i o n such that F =[x (0) ,...,x (k) ], recall the subset of I(n +1,r 1) given by

I (n +1,r 1) . = {α 2I(n +1,r 1) : ↵ i =0fori< (0)} ,
F 2∆ k (T ) {(F, α) 2 ∆ k (T ) ⇥I (n +1,r 1)} ,
whose cardinality coincides, by construction, with that of e J (n +1,r 1). We hence define

X k r,min (T ) . = n s (F,α) 2 X k r (T ):( F, α) 2 e I(n +1,r 1) o . (2.18)
The idea consists in killing a direction that can be obtained as "linear combination" of the others. To fix ideas, consider n =3a n dk =2 . W eh a v ef o u r 2-faces F 2 ∆ 2 (T ), so any vector parallel to the 2-face F 0 can be written as linear combination of vectors parallel to F i ,w i t hi =1 , 2, 3. This motivates the choice in (2.18). For ease of construction it is convenient to relate such a direction with an increasing numeration of the vertices of T . Remark 2.26. For r =1 , X k r,min (T ) and X k r (T ) coincide for each k. In fact, I(n +1, 0) = ; implies that e I(n +1, 0) = ; as well. For k =0, since one has that X 0 r,min (T )=L r (T ), by Lemma 2.11 we have that X 0 r,min (T ) coincides with X 0 r (T ). For k = n we have that

X n r,min (T )=X n r (T ).
Indeed, when k = n the only increasing permutation is the identity id : n ! n, so there is a trivial bijection between the sets e I(n +1,r 1) = (id, I(n +1,r 1)) and I(n +1,r 1) itself. We hence do not remove any element from X n r (T ).

Matching (2.18) with (2.16) we see that #X k r,min (T )=# Σ k r,min (T ). Hence, if we prove that X k r,min (T )i su n i s o l v e n tf o rP r Λ k (T ), then it is also minimal.

Lemma 2.27. Let F be a k-simplex supported in T . The set X k r,min (F ) is unisolvent for P r Λ k (F ).

Proof. Since F is a k-simplex, X k r,min (F )=X k r (F ),
(T )=X k 1 (T )=∆ 1 (T ). Assume then that r>1a n d0<k<n .W es h o wt h a ti f! 2P r Λ k (T )i ss u c h that R s ! =0foralls 2 X k r,min (T )t h e n R s 0 ! =0foralls 0 2 Σ k r,min (
T ). The result will thus follow from Proposition 2.25.

Assume

! 2P r Λ k (T )i ss u c ht h a t R s ! =0f o re a c hs 2 X k r (F ), F 2 ∆ k (T ), and observe that X k r (F )=X k r,min (T ) \ F since F 2 ∆ k (T )
. By replacing T with F in Proposition 2.17, it follows that ! F =0,hence R s 0 ! =0foran yk-simplex s 0 supported in F .I np a r t i c u l a r Z

s 0 ! =0 foreac hs 0 2 Σ k r,min (T ) \ F, F 2 ∆ k (T ). ( 2

.19)

Each F = F 2 ∆ k (T )i sa s s o c i a t e dw i t hap e r m u t a t i o n : k ! n.F o ra n y (i.e. for any F = F ), consider the set ¯ = n ,i . e . t h ec o m p l e m e n to f in n. Then, for each 6 =idrepresen tingsomeF 2 ∆ k (T )a n dα 2I (n +1,j), put for all j<r It is worth noting the concrete meaning of (2.20), represented in Figure 2.12, since those objects will be used and strongly simplified later.

⇡ ,α . = n x 2 T : i (x)= ↵ i r , 8i 2 ¯ o . (2.20) Since 6 =id,X k r,min (T ) \ ⇡ ,α = X k r (T ) \ ⇡ ,α = X k r j (⇡ ,α ). Thus the hypothesis yields Z s ! =0 foreac hs 2 X k r j (⇡ ,α ). ( 2
x 0 x 1 x 2 x 3 x 0 x 1 x 2 x 3 Figure 2
.12: There is only one 2-simplex parallel to [x 0 , x 1 , x 2 ]a tafi x e dd i s t a n c e (left) hence ⇡ i,j (shaded blue triangle) is well defined, but there are two 1-simplices at a fixed distance that are parallel to [x 0 , x 2 ], which belong to ⇡ ,α with |α| = j (right, red segments).

Some applications

Explicit computations for k =1and k = n 1

The main concern of the above approach is that n and k are generic, so no evident simplifications can be carried.

We present two applications of the ab ove results for what concerns the case of k =1a n dk = n 1. The former case is deepened in [1] and, for the particular case of k =2a n dn = 3, the latter represents the main result of [3]. In both cases we here develop proofs contained in those papers. Although those articles contain specific cases, they were a strong motivation for this whole section, since they present the fundamental techniques that we shall need for the forthcoming results. We thus reinterpret [1] and [2] as direct consequences of this section.

The case k =1

The case for k =1a n dn = 2 is interesting and easy to visualise and is depicted graphically in Figure 2.13. It first appeared in [1]. Such a case, which might seem rather restrictive, on the contrary not only does suggest the deep connection that links the space Σ k r,min (T )andX k r,min (T ), but may also be carried in a very evident fashion for k = 1 in any dimension n 1.

x 0 x 1 x 2 Figure 2
.13: Elements of Σ 1 2,min (T )o ne a c he d g e[ x i , x j ] may be obtained by chopping: for instance, the second half of [x 1 , x 2 ]i so b t a i n e db ys u b t r a c t i n gt h e red small edge from the long blue one. On the other hand, this is the only way to write such a small edge.

In general, in this case the linearity of integral comes to the rescue and yields aw i d em e t h od o l o g y ,w h i c hw ec a l lchopping.

Theorem 2.29 (The chopping strategy). Let k =1. The set X 1 r,min (T ) is unisolvent for P r Λ 1 (T ).

Proof. Let ! 2P r Λ 1 (T ). We shall prove that if Z s ! =0 8s 2 X 1 r,min (T ) then ! =0 . T od os ow es h o wt h a ti f R s ! =0f o re a c hs 2 X 1 r,min (T ), then R s ! =0f o re a c hs 2 Σ 1 r,min (
T ). Hence the result will follow from the case k =1 of Proposition 2.25.

Recall that any element of X 1 r,min (T )m a yb ew r i t t e na si n( 2 . 1 8 )a n da n y element of Σ 1 r,min (T )a si n( 2 . 1 6 ) . E x p l i c i t l y ,w eh a v et h a t

X 1 r,min (T ) . = n s X (F,α) 2 X 1 r (T ):( F, α) 2 e I(n +1,r 1) o and Σ 1 r,min (T ) . = n s Σ (F,θ) 2 Σ 1 r (T ):( F, θ) 2 e J (n +1,r 1) o , with s X (F,α) = 1 r 0 B @ F +[x 0 | ...|x n ] 0 B @ ↵ 0 . . . ↵ n 1 C A 1 C A and s Σ (F,θ) = ✓ 0 +1 r 0 B @ F +[x 1 | ...|x n ] 0 B @ ✓ 1 . . . ✓ n 1 C A 1 C A .
We write any element of Σ 1 r,min (T )a st h ef o r m a ls u mo fe d g e si nX 1 r,min (T ). Since F = F 2 ∆ 1 (T )i sa ne d g e ,i ti sr e p r e s e n t e db yap e r m u t a t i o n =[ (0), ( 1)]. Put ¯ = n .T h e ns X (F,α) and s Σ (F,θ) belong to the same line if and only if ↵ i = ✓ i for each i 2 ¯ .D e fi n e α ¯ ,j . = ↵ 0 ,...,↵ (0) j, . . . , ↵ (1) + j, . . . , ↵ n .

For any α,f o rj =0,...,✓ 0 ,w et h u sw r i t ea n ys Σ (F,θ) 2 Σ 1 r,min (T )a s

s Σ (F,θ) = ✓ 0 [ j=0 s X (F,α σ,j ) . (2.22)
By the linearity of integral we get

Z s Σ (F,θ) ! = Z S θ 0 j=0 s X (F,α σ,j ) ! = ✓ 0 X j=0 Z s X (F,α σ,j ) !,
which thus yields the relationship

Z s ! =0 8s 2 X 1 r,min (T )= ) Z s ! =0 8s 2 Σ 1 r,min (T ).
By Proposition 2.25 we get the claim. Observe that the converse holds true as well, by reversing (2.22) and writing elements of X 1 r,min (T )a sd i ff e r e n c eo fs e g m e n t si n Σ 1 r,min (T ). Equation (2.22) motivates the name chopping.I nf a c t ,o n em a yt h i n ko ft h a t splitting as considering a whole edge (which is an element of Σ 1 r,min (T )) and break it into smaller pieces, as one does with a wooden stick, whence the name. Similarly, the linearity of the integral represents the action of sticking together the ends of

x 1 x 2 x 0 s Σ (F 0 ,(2,0,0)) s X (F 0 ,(0,2,0)) s X (F 0 ,(0,1,1))
s X (F 0 ,(0,0,2))

Figure 2.14: Visual depiction of chopping.F o l l o w i n g( 2 . 2 2 ) ,o n eh a ss Σ (F 0 ,(2,0,0)) = s X (F 0 ,(0,2,0)) [ s X (F 0 ,(0,1,1)) [ s X (F 0 ,(0,0,2)) .

adjacent pieces of wood. This is clearly a peculiarity of the case k =1 . I ti s depicted in Figure 2.14. The strategy of chopping makes it possible to understand explicitly why some weights are linearly dependents from others. It also says which weights are dependent on which others. The following examples clarifies this fact.

Example 2.30. With respect to Figure 2.13, we show that the integrals associated with elements of Σ 1 2 (T ) lying on [x 1 , x 2 ] are linearly dependent. We show that the redundant rows associated with elements of [x 1 , x 2 ] are linear combination only of rows associated with small simplices supported in [x 1 , x 2 ] itself. Consider thus ! = p( 0 , 1 , 2 )d 1 + q( 0 , 1 , 2 )d 2 . Since we aim to compute integrals over [x 1 , x 2 ], we observe that ! [x 1 ,x 2 ] = e p( 1 )d 1 , being e p a degree 2 polynomial, say e p( 1 )=a 2 1

+ b 1 + c. Call then m the midpoint of [x 1 , x 2 ]. We have: 8 > < > : R m x 1 ! = 1 24 a + 1 8 b + 1 2 c R x 2 x 1 ! = 1 3 a + 1 2 b + c R x 2 m ! = 7 24 a + 3 8 b + 1 2 c
whence one easily gets that the third integral is the second minus the first, which was expected since

[x 1 , x 2 ]=[ x 1 , m] [ [m, x 2 ]
. Notice that this is independent from the fact that m is the midpoint.

The case k = n 1

To motivate this construction we first consider the visualisable case of a 3-simplex T ⇢ R 3 .W eg i v ead i r e c tp r o o fo fT h e o r e m2 . 2 8i nt h ef a s h i o nk =2a n dn =3 , which has a neat outline.

Proposition 2.31 ([2], Proposition 4). Let ! 2P r Λ 2 (T ) be such that Z s ! =0 8s 2 X 2 r,min (T ).
Then ! =0.

Proof. In fact, we show that if

! 2P r Λ 2 (T )i ss u c ht h a t R s ! =0f o ra l ls 2 X 2
r,min (T )t h e n R s 0 ! =0f o ra l ls 0 2 Σ 2 r,min (T ). The result will thus follow from Proposition 2.25.

Assume

! 2P r Λ 2 (T )issuchthat R s ! =0foreac hs 2 X 2 r (F )andF 2 ∆ 2 (T )
. By replacing T with F in Proposition 2.17, it follows that

R s 0 ! =0f o ra n y2 - simplex s 0 supported in F . In particular Z s 0 ! =0 foreac hs 0 2 Σ 2 r,min (T ) \ F, F 2 ∆ 2 (T ). (2.23)
For each face F i = T [x i ]w i t hi 2{ 1, 2, 3} and j = {1,...,r 1} consider ⇡ i,j . = {x 2 T : i (x)= j r }.S i n c e i 6 =0 ,X 2 r,min (T ) \ ⇡ i,j = X 2 r (T ) \ ⇡ i,j = X 2 r j (⇡ i,j ). Thus the hypothesis yields Z s ! =0 foreac hs 2 X 2 r j (⇡ i,j ).

(2.24)

Since !| F i =0fori 2{1, 2, 3},then!| ⇡ i,1 2P r 1 Λ 2 (⇡ i,1
), hence by Proposition 2.17 we conclude that !| ⇡ i,1 = 0. As a consequence,

R s 0 ! =0f o re a c hs 0 2 Σ 2 r 1,min (T ) \ ⇡ i,1 .
O n ec a nc o n t i n u ei nt h i sw a yf o rj =2,...,r 1todeducewith the same argument that !| ⇡ i,j =0 ,h e n c e R s 0 ! =0f o re a c hs 0 2 Σ 2 r,min (T ) \ ⇡ i,j , for i 2{1, 2, 3} and j 2{1,...,r 1}.T o g e t h e rw i t h( 2 . 2 3 )w eh a v e R s 0 ! =0for each s 0 2 Σ 2 r,min (T ), hence one has ! =0b yProposition2.25.

The above result was presented in [2] in the explicit fashion of face elements. However, we may notice that in the proof of Proposition 2.31 we have just used the fact that k = n 1a n dn o ts p e c i fi c a l l yt h a tn =3a n dk =2 . I n d e e d ,i n such a case the set ⇡ ,α always reduces to a way easier structure, since = i and α = |α| = j, see Figure 2.12. We may thus generalise it to any n as follows.

Corollary 2.32 (of Theorem 2.28). Let T be an n-simplex. Let ! 2P r Λ n 1 (T ) be such that Z s ! =0 8s 2 X n 1 r,min (T ).

Then ! =0.

Proof. The proof of this result is very close to that of Proposition 2.31, with n 1i np l a c eo f2 . I n d e e ds 2 X n 1 r,min (T )i sp a r a m e t r i s e db yap a i r( α,F)w i t h F 2 ∆ n 1 (T ). Thus F is associated with a permutation : n 1 ! n,h e n c e ¯ = n = i for some i =0,...,n.I tf o l l o w st h a ta n y⇡ ,α reduces to a portion of hyperplane depending only on two parameters ⇡ i,j . = {x 2 T : i (x)= j r },with j =0,...,r 1.

We show that if

! 2P r Λ n 1 (T )i ss u c ht h a t Z s ! =0 8s 2 X n 1 r,min (T ), then Z s ! =0 8s 2 Σ n 1 r,min (T ),
whence we get the result.

Assume ! 2P r Λ n 1 (T )i ss u c ht h a t R s ! =0f o re a c hs 2 X n 1 r (F )a n d F 2 ∆ n 1 (T )
. By replacing T with F in Proposition 2.17, it follows that R s 0 ! =0 for any (n 1)-simplex s 0 supported in F .I np a r t i c u l a r Z

s 0 ! =0 foreac hs 0 2 Σ n 1 r,min (T ) \ F, F 2 ∆ n 1 (T ). ( 2

.25)

For each (n 1)-face F i = T [x i ]w i t hi 2{1,...,n} and j = {1,...,r 1} consider the portion of hyperplane ⇡ i,j . = {x 2 T : i (x)= j r }.S i n c ei 6 =0 ,

X n 1 r,min (T ) \ ⇡ i,j = X n 1 r (T ) \ ⇡ i,j = X n 1 r j (⇡ i,j ). Thus the hypothesis yields Z s ! =0 foreac hs 2 X n 1 r j (⇡ i,j ). (2.26) Since !| F i =0f o ri 2{ 1,...,n},t h e n!| ⇡ i,1 2P r 1 Λ n 1 (⇡ i,1
), hence by Proposition 2.17 we conclude that

!| ⇡ i,1 = 0. As a consequence, R s 0 ! =0f o r each s 0 2 Σ n 1 r 1,min (T ) \ ⇡ i,1 .
O n ec a nc o n t i n u ei nt h i sw a yf o rj =2 ,...,r 1 to deduce with the same argument that !| ⇡ i,j =0 ,h e n c e R s 0 ! =0f o re a c h s 0 2 Σ n 1 r,min (T ) \ ⇡ i,j ,f o ri 2{ 1,...,n} and j 2{ 1,...,r 1}.T o g e t h e r w i t h (2.25) we have R s 0 ! =0foreac hs 0 2 Σ n 1 r,min (T ), hence one has ! =0b yProposition 2.25.

Chapter 3 Non uniform high order weights

In Chapter 2 we have dealt with unisolvence results that hold for every dimension n, all order of k-forms and any polynomial degree r.T od os ow eh a dt oi m p o s es o m e strict constraints on the geometry of the small simplices we considered, confining ourselves to uniformity. In practical situations, very often one may encounter the necessity of deducing the unisolvence of weights that are associated with domains that do not boil down to such simplifications. Also, in applications one does not really need such a vast (in fact, infinite!) amount of polynomial degrees, and due to computational costs and possible instability one usually reduces to deal with polynomials that barely overcome degree 5 or 6. We provide then a computational tool, which we call generalised Vandermonde matrix,t h a ta l l o w st oc h e c kw h e t h e r as p e c i fi cc o l l e c t i o no fw e i g h t si su n i s o l v e n tf o rP r Λ k (T ). This matrix represents apairingbet w eenabasisofP r Λ k (T ) and a collection of simplices via integration: each column of this matrix is associated with an element of a basis of the space of polynomial differential forms and each row is associated with a different simplex. We give a characterisation showing that unisolvent and minimal collections of simplices offer an invertible generalised Vandermonde matrix V .T h u st h ei nvertibility of V consists in asking that the resulting collection of weights is a basis for P r Λ k (T ) ⇤ .I n l i t e r a t u r e [ 5 9 ] [ 6 0 ] a u t h o r s u s u a l l y d e a l w i t h a n o t h e r , n o n invertible matrix, which allows to check unisolvence but not minimality. It represents the same pairing (in fact, its transpose), but instead of considering a basis for P r Λ k (T ) ⇤ ,itisconstructedusingasystemofgeneratorsassociatedwiththe totality of small simplices. In such a case, that existence of N =d i mP r Λ k (T ) linearly independent columns is the desired condition for unisolvence and the corresponding small simplices gives thus also a minimal collection. Note that there may exist more than a single choice of N linearly independent columns. The invertibility of the generalised Vandermonde matrix does not depend on the basis chosen for P r Λ k (T ). Once the invertibility of V is achieved, we may thus look for a preferred basis. By taking the inverse transpose of the generalised Vandermonde matrix we then extend the concept of cardinal basis to this context. In order to ensure a correct inversion, we thus look for a basis which is both easy to implement and that reduces the conditioning of such a matrix. For this reason the Bernstein basis is introduced.

This construction opens the way to the definition of the generalised Lebesgue constant,at o o lt h a tm e a s u r e st h eq u a l i t yo faf a m i l yo fw e i g h t s . I te x t e n d st h e concept of Lebesgue constant used in nodal interpolation. The relevance of this topic consists in fact in allowing techniques of nodal interpolation to study the distribution of simplices inside the simplex. We shall see though that it presents some differences with respect to the nodal Lebesgue constant: for instance, it does depend on the domain and not only on the placement of simplices inside the element. We thus dedicate a section of this chapter to studying the relationship of the generalised Lebesgue constant with the domain.

In order to compare meaningful families of weights a technique for generating new collections of simplices is needed. Simplicial isomorphisms come into play and are adopted to deform the set of small simplices. The strategy we present consists in considering maps that move points of a principal lattice to nodes that are well suited for nodal interpolation. New k-simplices are then built joining these points following the connectivity inherited from small simplices. This choice also ensures minimality. As soon as one moves away from the uniform configuration, theoretical results of Chapter 2 fail, so the invertibility of the generalised Vandermonde matrix is invoked to check unisolvence of the so obtained sets. A particular attention is paid to the condition number of such matrices. We conclude this chapter by providing an explicit construction based on warp and blend mappings and checking that we indeed keep unisolvence. Reasons that lead to such a map are multiple: it depends on a parameter which thus makes it easy to move nodes. More specifically, for appropriate choices of this parameter one is able to mimic widely studied collection of nodes, such as symmetrised Lobatto and Fekete points. This allows for comparisons with nodal results known in literature. This will be exploited in Chapter 4.

The generalised Vandermonde matrix

Definition 3.1 (Generalised Vandermonde matrix). Let {! 1 ,...,! N } be a basis for P r Λ k (T ). Let S k r (T ) be a family of k-simplices supported in T and fix an ordering {s 1 ,...,s N } for its elements. We define the generalised Vandermonde matrix associated with {! 1 ,...,! N } and {s 1 ,...,s N } as the square matrix V such that its (i, j)-th element is

V i,j = Z s i ! j . (3.1)
Thinking of S k r (T )a sav e c t o rs p a c ew h o s ee l e m e n t sa r ef o r m a lc o m b i n a t i o n s of k-simplices, the generalised Vandermonde matrix V represents the bilinear form

Z : S k r (T ) ⇥P r Λ k (T ) ! R (s, !) 7 ! Z s ! (3.2)
that pairs a family of small k-simplices and the space P r Λ k (T )a n dm a k e si t possible to treat the issue of unisolvence as a matter of linear algebra. Setting v as the coordinate vector of s in S k r (T )w i t hr e s p e c tt o{s 1 ,...,s N } and w as the coordinate vector of ! in P r Λ k (T )w i t hr e s pe c tt o{! 1 ,...,! N },t h e nw ec a n write (3.2) as v T V w.

Ac o n c e p to fg e n e r a l i s e dV a n d e r m o n d em a t r i xa l r e a d ya p pe a r si nl i t e r a t u r ea s ap a i r i n gbe t w e e n( ab a s i so f)t h es p a c eo fpo l y n o m i a l s( o rpo l y n o m i a ld i ff e r e n t i a l forms) and the corresponding degrees of freedom; see, for instance, [32, Section 5.2]. Definition 3.1 is thus coherent with it.

Proposition 3.2. The set of weights associated with the collection of k-simplices S k r (T ) is unisolvent and minimal for P r Λ k (T ) if and only if the generalised Vandermonde matrix V is invertible.

Proof. Order the simplices of S k r (T )a s{s 1 ,...,s N } and fix a basis {! 1 ,...,! N } for P r Λ k (T ), where N =dimP r Λ k (T ). Recall that

V i,j = Z s i ! j . Any ! 2P r Λ k (T )i se x p a n d e da s ! = N X j=1 a j ! j for a j 2 R. Hence Z s i ! = Z s i N X j=1 a j ! j = N X j=1 a j Z s i ! j . (3.4) Define u . = 0 B @ R s 1 ! . . . R s N ! 1 C A 57 Chapter 3
Weights as dofs for high order Whitney forms and expand its elements as in (3.4) to get

u = 0 B @ R s 1 ! 1 ... R s 1 ! N . . . . . . . . . R s N ! 1 ... R s N ! N 1 C A | {z } V 0 B @ a 1 . . . a N 1 C A | {z } a . (3.5)
Thus (3.5) is a linear system V a where the matrix of the coefficients is the generalised Vandermonde matrix. Hence V a implies that a =0i fa n do n l yi f ker V = {0},i . e . w h e nd e tV 6 =0.

The invertibility of the generalised Vandermonde matrix does not depend on the choice of the basis for P r Λ k (T ). In fact, write (3.2) in the matrix form (3.3) as v T V w, being V the generalised Vandermonde matrix written with respect to the given bases for S k r (T )a n dP r Λ k (T ). If we consider another basis {! 0 1 ,...,! 0 N } for P r Λ k (T ), then we can write

! 0 j = N X k=1 P j,k ! k
for some invertible matrix P .L e t V 0 be the generalised Vandermonde matrix written with respect to {! 0 1 ,...,! 0 N }.W eh a v et h a t ,b yt h el i n e a r i t yo ft h ei n t e g r a l ,

V 0 i,j . = Z s i ! 0 j = Z s i N X k=1 P j,k ! k = N X k=1 P j,k Z s i ! k = N X k=1 P j,k V i,k , so that V 0 = VP T .
In the matrix form, if we call w 0 the coordinate vector with respect to the basis

{! 0 1 ,...,! 0 N },t h i sb a s ec h a n g ebe c o m e s v T VP T w 0 = v T V 0 w 0 , (3.6)
where P T appears since coordinates are elements of P r Λ k (T ) ⇤ and so they change with the transpose. Since P is invertible, det V 6 =0i fa n do n l yi fd e tV 0 6 =0 . W ea r et h u si na position to choose a preferred basis for P r Λ k (T ), which is a generalisation of the Lagrange basis in nodal interpolation.

Definition 3.3 (Cardinal basis)

. Let S k r (T )={s 1 ,...s N } be a collection of small simplices that gives unisolvent and minimal weights for P r Λ k (T ). The cardinal (or dual) basis associated with

S k r (T ) is {! s 1 ,...,! s N } such that Z s i ! s j = i,j , (3.7)
being i,j the Kronecker symbol.

The cardinal basis functions are also known in finite elements as shape functions. Notice that if we choose a basis for P r Λ k (T )t h a ts a t i s fi e s( 3 . 7 ) ,t h e nt h e associated generalised Vandermonde matrix is the identity matrix. Thus, if V is invertible, we may consider Equation (3.6) and take P T = V 1 .I tf o l l o w st h a t once one is given the matrix V written with respect to a basis {! 1 ,...,! N } one may use it to explicitly compute the cardinal basis by 0

B @ ! s 1 . . . ! s N 1 C A = V T 0 B @ ! 1 . . . ! N 1 C A .
More explicitly, taking the i-th row of the above identity, we have

! s i = N X j=1 V T i,j ! j . (3.8)
If we drop the hypothesis of minimality we may still offer a handy version of Proposition 3.2.

Corollary 3.4. Let S k r (T ) be a set of k-simplices of cardinality M>N= dim P r Λ k (T ). Then S k r (T ) yields unisolvent weights for P r Λ k (T ) if and only if the matrix

W = 0 B @ R s 1 ! 1 ... R s 1 ! N . . . . . . . . . R s M ! 1 ... R s M ! N 1 C A has rank N .
Proof. It suffices to perform linear combinations on the rows of W to eliminate the redundant M N rows. It is not restrictive to suppose in particular that they are the last M N .O n et h e na p p l i e sP r o p o s i t i o n3 . 2t ot h er e s u l t i n gs y s t e m .

The matrix W appears when the hypothesis of minimality on the set S k r (T )i s dropped and one hence looks at a system of generators for weights in place of a basis. Thus, Corollary 3.4 may be exploited to identify such a set: in particular, it is sufficient to extract N elements of S k r (T )a s s o c i a t e dw i t hr o w st h a tm a k eW full rank. The corresponding simplices will then give a basis for weights. Remark 3.5. In literature, in particular in [START_REF]High order edge elements on simplicial meshes[END_REF] and [START_REF] Rapetti | Whitney forms of higher degree[END_REF], the matrix W is sometimes presented in a slightly different form. It also contains columns associated with polynomials discarded in Theorem 1.19 to obtain a basis for P r Λ k (T ) from a system of generators for that space. It is thus a square, degenerate matrix. In such a case one shall look for a minor of rank N to obtain unisolvence, as a consequence of Proposition 3.2 applied to such a submatrix.

Exact computation of the generalised Vandermonde matrix

Interestingly, (3.1) does not require a quadrature rule to be computed. This is basically due to the Lasserre-Avrachenkov Theorem [START_REF] Lasserre | The multi-dimensional version of R b a x p dx[END_REF], which states that the integral of a polynomial over a simplex T can be read in terms of evaluation of homogeneous forms on the vertices of T .T h ec o m p l e x i t yo ft h i st a s ki se x p l a i n e d in details in [START_REF] Baldoni | How to integrate a polynomial over a simplex,M a t h e m a t i c s of computation[END_REF]. However, exploiting the peculiarity of the space P r Λ k (T )a n d choosing an appropriate monomial basis of high order Whitney forms, an interesting simplification can be carried out. We follow and retrace [21, Proposition 3.15] and related results.

Let T =[ x 0 ,...,x n ] ✓ R n an n-simplex. Let F 2 ∆ k (T )
and suppose that F is associated with the increasing permutation

: k ! n,i . e . F = F = [x (0) ,...,x (k) ]. Put λ α . = ↵ σ(0) (0) ... ↵ σ(k)
(k) and ! F as the Whitney form associated with F .O nF ,w eh a v e( s e e [ 3 6 ]

o r[ 6 3 ] ) Z F λ α ! F = ↵ (0) ! ...↵ (k) !k! ↵ (0) + ...+ ↵ (k) + k ! Z F ! F , (3.9)
which reduces further recalling that R F ! F = 1 k! , see Lemma 1.14. Notice that this formula works when we integrate on F an expression written in barycentric coordinates taken with respect to F ,a si n[ 3 3 ] . I n t e g r a t i o no na n o t h e rs i m p l e x requires some work.

Consider another k-simplex F 0 =[y 0 ,...,y k ]s u p po r t e di nT .W ec o m p u t e Z

F 0 λ α ! F . (3.10)
Our aim consists in reducing Equation (3.10) to a sum of elements as in (3.9). To begin with, we observe that the restriction of ! F to F 0 is constant. In fact, if we call ' : F 0 ! F the affinity that maps a k-simplex to the other, we have that ! F F 0 = ' ⇤ ! F .T h el a t t e ri sa ne l e m e n to fP 1 Λ k (F 0 ), since the spaces of Whitney forms are invariant under the pullback with respect to affine transformations, as proved in Proposition 1.11. Thus ! F F 0 is constant on F 0 in view of Lemma 1.14. Now, since F 0 =[y 0 ,...,y k ]isak-simplex, we may define barycentric coordinates { 0 0 ,..., 0 k } with respect to its vertices. We may thus read { 0 0 ,..., 0 k } in terms of the k +1 vertices of F .T h u sc o n s i d e rt h em a t r i xM such that

M i,j . = (i 1) 1 y j 1 , (3.11)
which is a (k+1)⇥(k+1) invertible matrix since both F and F 0 are non degenerate k-simplices. We are assuming a shift in the indices of the matrix M ,w h i c hi sd u e to the fact that we run vertices and corresponding barycentric coordinates from 0 to k,whereasthematrixisindexedfrom1tok +1; thus M i,j should be thought of as the element in position i, j, that corresponds to the i-th barycentric coordinate and the j-th vertex in our enumeration, which is i 1 (y j 1 ). Notice that we may not read points outside (the k-surface containing) F 0 in terms of the coordinates 0 , but there we have, by construction, an explicit relationship with coordinates (i) that reads as

(i 1) = k+1 X j=1 M i+1,j 0 j 1 . (3.12)
Plugging this in the definition of ! F and manipulating, we get [21, Equation ( 85)], which is Z

F 0 ! F =detM. ( 3.13)
We now have all the ingredients to co ok Equation (3.10). Plugging (3.12) in, we obtain an integral on F 0 with respect to barycentric coordinates 0 .W ei n d e e dg e t Z

F 0 λ α ! F = Z F 0 ↵ σ(0) (0) ... ↵ σ(k) (k) ! F = Z F 0 k+1 X j=1 M 1,j 0 j 1 ! ↵ 0 ... k+1 X j=1 M k+1,j 0 j 1 ! ↵ k ! F .
Each of the terms in the latter integral splits as in Equation (3.9), and the remaining term R F 0 ! 0 ,w h i c ha p p e a r si ne v e r ys u m m a n d ,i ss o l v e db y( 3 . 1 3 ) . Ad e t a i l e d construction in T ⇢ R 3 for the elements of P 1 Λ k (T )i sc a r r i e di n[ 5 8 ] .

Example 3.6. We show how the above reasoning may be applied to the computation of weights associated with small simplices X k r (T ). To fix ideas, we assume k = n =2and r =3. The monomial basis for P r Λ 2 (T ) is thus

↵ 0 0 ↵ 1 1 ↵ 2 2 d 1 ^d 2 , with ↵ 0 + ↵ 1 + ↵ 2 =2.
Denote by M `the matrix that relates the barycentric coordinates of the simplex T and the `-th small simplex T `, as in (3.11); see Figure 3.1 for the relative

x 0 x 1 x 2 T 1 T 2 T 3 T 4 T 5 T 6
Figure 3.1: The 2-simplex T read with respect to its barycentric coordinates.

placement of T i in T . We compute

M 1 = 0 @ 1 2 3 2 3 0 1 3 0 00 1 3 1 A M 2 = 0 @ 2 3 1 3 1 3 1 3 2 3 1 3 00 1 3 1 A M 3 = 0 @ 00 1 2 3 00 1 3 1 A M 4 = 0 @ 2 3 1 3 1 3 0 1 3 0 1 3 1 3 2 3 1 A M 5 = 0 @ 1 3 00
1 3 2 3 1 3 1 3 1 3 2 3 1 A M 6 = 0 @ 00 1 3 0 2 3 1 1 A
and plug this in (3.10) to obtain the generalised Vandermonde matrix V . As an example, we use the formula just provided to explicitly compute the integral of 0 1 2 d 1 ^d 2 on T 2 . As an example, we compute Z

T 2 λ α ! T for α =(1, 0, 1). We get 0 = 3 X j=1 M 2 0,j 0 j = 2 3 0 0 + 1 3 0 1 + 1 3 0 2 , 1 = 3 X j=1 M 2 1,j 0 j = 1 3 0 0 + 2 3 0 1 + 1 3 0 2 , 2 = 3 X j=1 M 2 2,j 0 j = 1 3 0 2 , Chapter 3 whence Z T 2 λ α ! T = Z T 2 ✓ 2 3 0 0 + 1 3 0 1 + 1 3 0 2 ◆✓ 1 3 0 2 ◆ ! T = Z T 2 ✓ 2 9
0 0 0 2 + 1 9 0 1 0 2 + 1 9 0 2 2 ◆ ! T .
Exploiting the linearity of the integral we obtain six summands on which formula (3.9) can be used, since barycentric coordinates 0 i refer to T 2 . Thus we get Z

T 2 λ α ! T = 2! 4! ✓ 2 9 1!1! + 1 9 1!1! + 1 9 2! ◆ Z T 2 ! T = 5 12 • 9 Z T 2 ! T .
The latter integral is solved by (3.13), which gives

Z T 2 ! T =detM 2 = 1 9
and the computation is completed. Continuing in this way for all multi-indices in I(3, 2) we obtain the second row of generalised Vandermonde matrix Reproducing the same reasoning for T `, `=1 ,...,6, one constructs V . A direct computation shows that it is invertible.

V 2,j = 1 9 ✓ 22 54 , 22
To establish unisolvence of a given set of small simplices, we thus first check that det V 6 = 0 to see if hypotheses of Proposition 3.2 are satisfied. If so, since we aim to invert the matrix V to compute the cardinal basis as in (3.8), we verify that the condition number of such a matrix is not too large. Table 3.1 and Table 3.2 report condition numbers of the generalised Vandermonde matrices associated with uniform small simplices X k r,min (T ). They are thus the numerical counterpart of Theorem 2.28 and Theorem 2.25 for n =2andn =3.

Once the set of small simplices is fixed, the condition number cond(V )s h o w s also how much the basis chosen for P r Λ k (T )i si l l -c o n d i t i o n e d . I np a r t i c u l a r , notice that the conditioning is equal to 1 when we consider the cardinal basis, since in such a case V = I.

As t r a t e g yt or e d u c es u c hac o n d i t i o nn u m b e rc o n s i s t si ni n t r o d u c i n gac o e fficient in front of the monomial basis λ α ! F . Recalling that α . =( ↵ 0 ,...,↵ n ), we define the Bernstein basis [START_REF] Lorentz | Bernstein polynomials[END_REF] as the monomial basis multiplied by an appropriate coefficient, which reads as

B α ! F . = r! ↵ 0 ! •••↵ n ! λ α ! F . (3.14)
Improvements with respect to this basis have been presented for the edge element case in [3] and are here summarised in Table 3.1 for what concerns n =2a n di n Table 3.2 for n =3 . R e s u l t sf o rk = 0, i.e. for the nodal case, are diffused in literature, see for instance [START_REF] Blyth | A Lobatto interpolation grid over the triangle[END_REF] or [START_REF] Luo | A Lobatto interpolation grid in the tetrahedron[END_REF]. Table 3.1: Condition numb er of the generalised Vandermonde matrix asso ciated with X k r,min (T )f o rat r i a n g l eT ⇢ R 2 with respect to the total polynomial degree r.W e c o m p a r e t h e m o n o m i a lb a s i s λ α ! F for P r Λ k (T )( l e f t ,s e c o n da n dt h i r d columns) with the Bernstein basis B α ! F (two rightmost columns).

λ α ! F B α ! F r k =1 k =2 k =1 k =2 1
λ α ! F B α ! F r k =1 k =2 k =3 k =1 k =2 k =3
1 1.0000 ⇥ 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 We close the section by p ointing out the following. Remark 3.7. Once the basis for P r Λ k (T ) and the relative position of simplices in S k r (T ) are fixed, the associated generalised Vandermonde matrix is global, i.e. it does not depend on the choice of the simplex T .

⇥

The generalised Lebesgue constant

The Lebesgue constant is a central instrument in nodal interpolation. It measures the stability of the interpolation with respect to the placement of interpolation nodes on the domain. In the nodal case, i.e. for k =0 ,i ti sd e fi n e da s( s e e ,f o r instance, [START_REF] Quarteroni | Numerical Mathematics[END_REF]Chapter 8])

Λ r =sup x2I N X i=1 |l i (x)| , ( 3
l i (x)= N Y j=1 j6 =i x p j p i p j .
Note that, as an alternative, one may use any basis of P r (I), compute the Vandermonde matrix and recover l i (x) by applying (3.8). The role of (3.15) in nodal interpolation is celebrated. In particular, it is well known that it appears as a measure of stability and as an indicator of quality of (local) interpolation [START_REF] Quarteroni | Numerical Mathematics[END_REF]Section 8.1]. The following two results, which we will extend to this framework in Chapter 4, set a motivation.

Let f : I ! R be a continuous function and let Π r f denote its interpolated in the space P r (I)o fp o l y n o m i a l so fd e g r e ea tm o s tr.S u p p o s et h a tw ea i mt o interpolate f knowing its evaluations at points {x 1 ,...,x N }.S u p p o s et h a tv a l u e s f (x i )maybeperturbed,sothat,inplaceoff ,w einfactkno wv aluesofafunction e f (x i ). This happens, for instance, when dealing with physical measurements. We may not exp ect to reconstruct exactly f provided these values, but via the Lebesgue constant we can bound the propagation of the error in the interpolated. In fact, one has

sup x2I |Π r f (x) Π r e f (x)|Λ r max i=1,...,N |f (x i ) e f (x i )|. (3.16)
Clearly the term |f (x i ) e f (x i )| appearing on the right hand side is finite though not in general known, but in application can be estimated, for instance, by the (3.17)

Again, if we are able to control the quantity Λ r ,w em a ye x p e c tab e t t e ra p p r o ximation. A famous example of what can happen when considering nodes that offer large Lebesgue constants is the Runge counterexample (see Figure 3.2). A common technique to overcome this problem consists in varying the interpolation points and consider non equidistributed nodes, such as Chebychev nodes [START_REF] Quarteroni | Numerical Mathematics[END_REF]. We will come back on this and provide a generalisation to this fact in Section 4.4. Equation (3.15) is easily understood also for the case of several variables [START_REF] Bos | Bounding the Lebesgue function for interpolation in a simplex[END_REF], although explicit formulas for l i (x):R n ! R are not as neat as that for one dimension. Again, the generalised Vandermonde matrix may then be used to compute such Lebesgue functions, and Λ r is then defined just by taking the supremum over the multi-dimensional domain. Explicit definitions for R 2 and R 3 have been given in [START_REF] Bloom | The Lebesgue constant for Lagrange interpolation in the simplex[END_REF]. Several works and examples for R 2 and R 3 have also been worked out, see for instance [START_REF] Blyth | A comparison of interpolation grids over the triangle or the tetrahedron[END_REF]. As in the one dimensional case, Lebesgue constant can be contained by abandoning uniform nodes for non uniform ones.

Very recently, in [START_REF] Alonso Rodríguez | On a generalization of the Lebesgue's constant[END_REF], this concept has been extended to Whitney forms by means of the generalised Vandermonde matrix. In particular, consider a family of unisolvent and minimal small simplices S k r (T )={s 1 ,...,s N } and the associated cardinal basis {! s 1 ,...,! s N },a si nD e fi n i t i o n3 . 3 . L e tC k (T )d e n o t et h es e to f k-chains supported in T .T h egeneralised Lebesgue function is

L : C k (T ) ! R L(c)= N X j=1 |s j | 0 Z c ! s j , (3.18)
where |s j | 0 is the k-th dimensional volume of s j (that is, its Hausdorff measure).

We thus have the following definition.

Definition 3.8 (Generalised Lebesgue constant). Let S k r (T )={s 1 ,...,s N } be a family of unisolvent and minimal small simplices and let {! s 1 ,...,! s N } be the associated cardinal basis. Let L(c) be the Lebesgue function. The generalised Lebesgue constant is

Λ r,k . =s u p c2C k (T ) 1 |c| 0 L(c)= sup c2C k (T ) 1 |c| 0 N X j=1 |s j | 0 Z c ! s j . (3.19)
Since the mass of a point c 2 R n is 1 and R c ! = !(c)w h e n! is a 0-form, Equation (3.19) boils down to (3.15) for k =0 ,t h a ti s ,Λ r,0 =Λ r .T h i s i s [ 5 , Proposition 1], and is also carried in details in [3].

Remark 3.9. The construction of the cardinal basis {! s 1 ,...,! s N } involved in the computation of (3.19) requires the inversion of the generalised Vandermonde matrix. To this end a basis for P r Λ k (T ) needs to be fixed, see Equation (3.8).W e show that Λ r does not depend in fact on this choice. Indeed, let

ω . = 0 B @ ! 1 . . . ! N 1 C A and η . = 0 B @ ⌘ 1 . . . ⌘ N 1 C A
be two bases for P r Λ k (T ). Let V and V 0 be the generalised Vandermonde matrices written with respect to {! 1 ,...,! N } and {⌘ 1 ,...,⌘ N }, respectively. We know that there exists an invertible matrix P such that η = P ω, whence ω = P 1 η. Call ). Now, let Λ r,k (T ' )a n dΛ r,k (T )d e n o t e ,r e s p e c t i v e l y ,t h eg e n e r a l i s e dL e b e s g u e constant associated with S k r (T ' )a n dt h a ta s s o c i a t e dw i t hS k r (T ). In order to relate them, we shall be able to understand the transformation of terms appearing in Equation (3.19) under the action of '. We thus work term by term. Lemma 3.11. Let {! '(s 1 ) ,...,! '(s N ) } be the cardinal basis associated with S k r (T ' ) and {! s 1 ,...,! s N } that associated with S k r (T ). The pullback induced by the affinity ' maps one into the other. Namely, for i =1,...,N, one has

ω s . = 0 B @ ! s 1 . . . ! s N 1 C A hence V ' is invertible if
! s i = ' ⇤ ! '(s i ) .
Proof. Recall that ! s i and ! '(s i ) are the only forms (respectively in P r Λ k (T )a n d

P r Λ k (T ' )) such that Z s j ! s i = i,j = Z '(s j ) ! '(s i ) .
We compute

i,j = Z '(s j ) ! '(s i ) = Z s j ' ⇤ ! '(s i ) . (3.20)
Hence Z

s j ' ⇤ ! '(s i ) =
( 1i f '(s j )='(s i ) , s j = s i 0o t h e r w i s e , so ' ⇤ ! '(s i ) satisfies the definition of ! s i ,w h e n c e' ⇤ ! '(s i ) = ! s i .

As an immediate consequence we also get the following, which adjusts the latter term of (3.19). Corollary 3.12. Let c be a k-simplex supported in T and '(c) the corresponding simplex supported in T ' . Then, for i =1,...,N, Z

'(c) ! '(s i ) = Z c ' ⇤ ! '(s i ) = Z c ! s i .
It is worth pointing out that, when k =0 ,c is a point and hence we have an equivalence of evaluations of functions of cardinal bases that reads as

! '(s i ) ('(c)) = ! s i (c). (3.21)
This will be used in Proposition 3.13. Now, the trickiest part consists in estimating how the k-volumes are transformed under the action of '.T h i sc l e a r l yd e p e n d so n l yo nt h em a t r i xM .I nf a c t , it turns out that an exact description is possible only in few cases. We premise them as a motivation for the resulting work. Proposition 3.13. Let ' : T ! T ' be a non degenerate affinity. One has Λ r,0 (T )=Λ r,0 (T ' ). (3.22) Proof. This descends from the fact that the mass of a point is 1 and Equation (3.21) of Corollary 3.12. In fact, we have

Λ r,0 (T ' )= sup c 0 2C 0 (Tϕ) 1 |c 0 | 0 N X j=1 |'(s j )| 0 Z c 0 ! '(s j ) =s u p '(c)2C 0 (Tϕ) N X j=1 ! '(s j ) ('(c)) =s u p c2C 0 (T ) N X j=1 ! s j (c) =s u p c2C 0 (T ) 1 |c| 0 N X j=1 |s j | 0 Z c ! s j =Λ r,0 (T ).
Taking the first and the last term we have the claim.

The above result states that the usual Lebesgue constant does not depend on the simplex T .T h ef o l l o w i n go n es h o w st h es a m ef o ra nn-simplex. This was also to be expected for the duality between 0-and n-forms in R n . Proposition 3.14. Let ' : T ! T ' be a non degenerate affinity. One has Λ r,n (T )=Λ r,n (T ' ).

(3.23)

Proof. We prove the assertion by direct computation. In such a case the volume of an n-simplex is transformed as |'(s j )| 0 = | det M ||s j | 0 .P l u g g i n ge v e r y t h i n gi n and exploiting Corollary 3.12, we have

Λ r,n (T ' )= sup c 0 2C k (Tϕ) 1 |c 0 | 0 X '(s j )2S n r (Tϕ) |'(s j )| 0 Z c 0 ! '(s j ) = sup '(c)2C k (Tϕ) 1 |'(c)| 0 X '(s j )2S n r (Tϕ) |'(s j )| 0 Z '(c) ! '(s j ) =s u p c2C k (T ) 1 | det M ||c| 0 X s j 2S n r (T ) | det M ||s j | 0 Z '(c) ! '(s j ) =s u p c2C k (T ) 1 |c| 0 X s j 2S n r (T ) |s j | 0 Z c ! s j =Λ r,n (T ).
Proof. Quantities | max | and | min | allow to bound the volume of T ' as shown in Lemma 3.15. Thus, applying both sides of (3.26), we have

Λ r,k (T ' )= sup c 0 2C k (Tϕ) 1 |c 0 | 0 X '(s j )2S k r (Tϕ) |'(s j )| 0 Z c 0 ! '(s j )  sup c 0 2C k (Tϕ) 1 |c 0 | 0 X s j 2S k r (T ) | max | k |s j | 0 Z c ! s j  sup c2C k (T ) 1 | min | k |c| 0 X s j 2S k r (T ) | max | k |s j | 0 Z c ! s j = | max | k | min | k sup c2C k (T ) 1 |c| 0 X s j 2S k r (T ) |s j | 0 Z c ! s j = ✓ | max | | min | ◆ k Λ r,k (T ).
This concludes the proof.

Obviously, since any diagonal matrix M is normal, i.e.

MM T = M T M ,t h e 2-condition number of M reads 2 (M )= | max| | min | ,w h e n c e Λ r,k (T ' )  k 2 (M )Λ r,k (T ). (3.28)
As we shall see with a numerical example in Section 4.2.1, result of Proposition 3.16 cannot be extended to diagonalisable matrices that are not already diagonal. On the contrary, if we drop assumptions on M we may still prove that (3.28) holds. As a motivation, we show the case k =1. T oseeit,w epremiseatec hnical lemma that evidences the role of kM k 2 in this context. It is a classical result and is adopted, in a similar way, in [3].

Lemma 3.17. Let e =[ p, q] be an edge supported in T and '(x)=M x + b be a non degenerate affinity. Denote by '(e) the edge supported on T ' generated by vertices '(p) and '(q). Then

|'(e)| 0 kM k 2 |e| 0 .
Proof. Since '(e)i sa ne d g e ,w eh a v e|'(e)| 0 = k'(q) '(p)k 2 .T h u s ,a p p l y i n g Cauchy-Schwarz inequality, we get

|'(e)| 0 = k'(q) '(p)k 2 = kM q + b M p bk 2 = kM (q p)k 2 kM k 2 kq pk 2 = kM k 2 |e| 0 .
Taking the first and the last term we get the claim.

We are in a p osition to prove (3.28) when k =1.

Proposition 3.18. Let '(x)=M x + b be a non degenerate affinity and let

T ' = '(T ). When k =1, one has Λ r,1 (T ' )  2 (M )Λ r,1 (T ). (3.29)
Proof. To see this, recall that 2 (M ) . = kM k 2 kM 1 k 2 .F o rk =1s i m p l i c e sa r e edges, hence |s j | 0 and |c| 0 are just Euclidean distances. Thus by Lemma 3.17 one has

|'(s j )|k M k 2 |s j |. Now, if c 0 = '(c)o n eh a sc = ' 1 (c 0 ), where ' 1 (x)= M 1 x b 0 ,b e i n gb 0 .
= M 1 b.T h i s i s a g a i n a n o n d e g e n e r a t e a ffi n i t y , h e n c e applying Lemma 3.17 once more one gets

|c 0 | 0 kM 1 k 1 2 |c| 0 .T h u s Λ r,1 (T ' )= sup c 0 2C 1 (Tϕ) 1 |c 0 | 0 X '(s j )2S 1 r (Tϕ) |'(s j )| 0 Z c 0 ! '(s j )  sup c 0 2C 1 (Tϕ) 1 |c 0 | 0 X s j 2S 1 r (T ) kM k 2 |s j | 0 Z c ! s j kM k 2 sup c2C 1 (T ) kM 1 k 2 |c| 0 X s j 2S 1 r (T ) |s j | 0 Z c ! s j = kM k 2 kM 1 k 2 Λ r,1 (T )= 2 (M )Λ r,1 (T ).
Hence, taking the first and the last term, we have

Λ r,1 (T ' )  2 (M )Λ r,1 (T ). (3.30)
The claim is proved.

Remark 3.19. Estimate (3.29) is interesting since it is universal, in the sense that it does not depend on any choice done on small simplices but involves only the matrix M representing the affinity '. Dropping this request, a sharper bound can be obtained. In fact, let s j . =[x j , y j ] be the edge of vertices x j and y j . Define

C(S 1 r ) . =max j M y j x j ky j x j k 2 2
, with j running over all the indices of small simplices s j . This bounds the stretching of the simplex s j , and by Cauchy-Schwarz inequality we immediately have that

C(S 1 r )  max j kM k 2 y j x j ky j x j k 2 2 = kM k 2 .
Thus, retracing the proof of Proposition 3.18 with this quantity, we get

Λ r,1 (T ' )  C(S 1 r )kM 1 k 2 Λ r,1 (T )  2 (M )Λ r,1 (T ).
Chapter 3

Before extending Proposition 3.18 to k>1, we shall adapt accordingly Lemma 3.17 to such a case. Lemma 3.20. Let T be a k-simplex and '(x)=M x + b be a non degenerate affinity. One has

|'(T )| 0 kM k k 2 |T | 0 . (3.31)
As a consequence, one also has

|'(T )| 0 kM 1 k k 2 |T | 0 . (3.32)
Proof. We prove (3.31) by induction. When k =1,T is an edge and this is Lemma 3.17. Now, assume k>1a n dt h a tt h ec l a i mh o l d sf o ra n y`=1,...,k 1. Fix a (k 1)-face F of T and call h the relative height, i.e. the distance between F and the opposite vertex, say v k .T h ev o l u m eo ft h ek-simplex T is

|T | 0 = 1 k |F | 0 h
and, likewise,

|'(T )| 0 = 1 k |'(F )| 0 h 0 , being h 0 the distance between '(F )a n d'(v k ). Now, call p the point in the (k 1)-plane containing F such that h =dist(v k , p)andcall'(q)thepoin tinthe (k 1)-plane containing '(F ) such that h 0 =dist('(v k ),' (q)
). Since '(p)lie so n this latter (k 1)-plane, one has

h 0 =dist('(v k ),'(q))  dist('(v k ),'(p)) kM k 2 h,
where last inequality is granted by Lemma 3.17 considering the edge joining v k and p and its image under '.

Applying this fact and the inductive hypothesis, which reads as

|'(F )| 0 kM k k 1 2 |F | 0 , since F is a (k 1)-simplex, we get |'(T )| 0 = 1 k |'(F )| 0 h 0  1 k kM k k 1 2 |F | 0 h 0  1 k kM k k 1 2 |F | 0 kM k 2 h = kM k k 2 1 k |F | 0 h = kM k k 2 |T | 0 .
To prove (3.32) we reproduce the same reasoning. For ease, put T ' . = '(T ), so that T = ' 1 (T ' ). Explicitly, we have ' 1 (x)=M 1 x b 0 ,be i n gb 0 = M 1 b.

We thus get

|T | 0 = |' 1 (T ' )| 0 kM 1 k k 2 |T ' | 0 = kM 1 k k 2 |'(T )| 0 , whence one retrieves immediately |'(T )| 0 kM 1 k k 2 |T | 0 .
We are in a p osition to prove the following.

Proposition 3.21. Let '(x)=M x + b be a non degenerate affinity and let

T ' = '(T ). One has Λ r,k (T ' )  k 2 (M )Λ r,k (T ). (3.33)
Proof. We carry out proof as in of Proposition 3.18. By direct computation and Lemma 3.20, we get

Λ r,k (T ' )= sup c 0 2C k (Tϕ) 1 |c 0 | 0 X '(s j )2S k r (Tϕ) |'(s j )| 0 Z c 0 ! '(s j )  sup c 0 2C k (Tϕ) 1 |c 0 | 0 X s j 2S k r (T ) kM k k 2 |s j | 0 Z c ! s j kM k k 2 sup c2C k (T ) 1 kM 1 k k 2 |c| 0 X s j 2S k r (T ) |s j | 0 Z c ! s j = kM k k 2 kM 1 k k 2 Λ r,k (T )= k 2 (M )Λ r,k (T ).
We have thus shown that Λ r,k (T

' )  k 2 (M )Λ r,k (T ).
We shall exhibit in Chapter 4 numerical results that verify the ab ove b ounds for different choices of weights. Also, we will present a case of a diagonalisable matrix M for which estimate (3.33) holds but (3.27) does not. This is Table 4.10.

From now on, if not differently specified, we will only consider the standard simplex T ⇢ R n as domain for the generalised Lebesgue constant, hence suppress the dependence. We will also drop the subscript k,whic hisimplicitintheorderof the form, so we simply denote it by Λ r ,i no r d e rt om a k et h ew o r km o r er e a d a b l e . We will study the role of this generalised Lebesgue constant in Chapter 4 and compute it for several choices of small simplices.

Simplicial isomorphisms

We may read the problem of unisolvence for high order Whitney forms as the generalisation of the problem of determining ap o l y n o m i a l ,i . e . t h ec a p a c i t yo f univocally reconstructing its coefficients from evaluations on appropriate nodes. The problem of unisolvence for differential k-forms deals then with the identification of collection of k-simplices that allows to define unisolvent (and possibly minimal) weights for such spaces of differential forms.

In Chapter 2 we treated the case of uniform nodes yielding uniform distributions. In that context, the word uniform was devoted to k-simplices whose vertices belong to the principal lattice L r (T ). Notice that this gave different choice of supports: for instance, both the sets X k r,min (T )a n dΣ k r,min (T )s a t i s f yt h i sr e q u e s t , although they are sensibly different (see, for instance, Figure 2.6 on page 37). Any other choice of nodes is referred to as nonuniform.

We now aim to weaken the constraint of uniformity. A p ossible approach consists in deforming existing unisolvent and minimal small k-simplices by moving their vertices to other well known distribution of nodes. In particular we shall consider nodes that are widely used in nodal interpolation. In this case only few theoretical considerations can be inherited from what we have already studied, but a great flexibility is gained.

We shall b orrow some concepts from discrete and differential geometry. To begin with, recall that a simplicial complex is a collection of simplices with appropriate gluing conditions. In particular, if S is a collection of simplices, they form a simplicial complex if and only if

• s 2 S implies that each s 0 2 ∆ k (s)be l o n g st oS as well;

• for each s, s 0 2 S,e i t h e rs \ s 0 = ; or s \ s 0 2 S.

We shall suppose that this collection is finite. An example of non trivial simplicial complex is depicted in Figure 1.2 on page 5.

For each r>0, we can collect spaces X k r (T )t oo b t a i nas i m p l i c i a lc o m p l e x , whose k-skeleton is precisely given by X k r (T ).

Definition 3.22 (Simplicial complex associated with small simplices). Let T ⇢ R n be an n-simplex. For each r>0 we define the simplicial complex associated with small simplices X k r (T ) as

X r (T ) . = n [ k=0 X k r (T ). (3.34)
We show that Definition 3.22 is well posed, i.e. that (3.34) in fact defines a simplicial complex. Lemma 3.23. For each r>0, X r (T ) has a structure of simplicial complex.

Proof. The fact that subsimplices of any s 2 X r (T )b e l o n gt oX r (T )t h e m s e l v e s is Lemma 2.9, since the boundary of any s 2

X k r (T )i st h ef o r m a lc o m b i n a t i o no f small (k 1)-simplices in X k 1 r (T ).
It remains only to show that two elements of X r (T ) either intersect on a common subsimplex that belongs to X r (T ) or do not intersect at all. We have already observed that two elements of X k r (T )i n t e r s e c ta tm o s ti nap o i n to ft h ep r i n c ipal lattice L r (T ). Since L r (T )=X 0 r (T )( s e ep r o o fo fL e m m a2 . 1 2 )t h ec l a i mi s proved.

For sure the simplicial complex X r (T ) contains unisolvent and minimal small n-simplices for P r Λ n (T ). For each k<n ,t h ek-skeleton of X r (T )i sau n i s o l v e n t (Proposition 2.17) but not minimal (Lemma 2.12) set for P r Λ k (T ).

The definition of a complex which contains minimal small k-simplices for each k is not possible. In fact, let us suppose that X r (T )isdefinedasin (3.34). Inview of Lemma 3.23, X r (T )containsalltheboundariesofX n r (T ), i.e.

S

s2X n r (T ) ∆ n 1 (s). The cardinality of this set is n+1 n times the number of small n-simplices, #X n r (T ), and is strictly greater than dim P r Λ n 1 (T ), as shown by Equation (2.7). On the contrary, if we defined X r (T )a st h eu n i o no fX k r,min (T ), we would not obtain a complex, as shown in Figure 3.3.

x 0 x 1 x 2 Figure 3.3:
In order to be a simplicial complex, the horizontal dashed edge must be included.

Maps between complexes are called simplicial maps. Among all these maps, we shall look for simplicial isomorphisms.C o n s i d e rt w oc o m p l e x e sX and Y and their vertex sets, respectively vert(X)a n dv e r t ( Y). Suppose that there exists a vertex map :v ert(X) ! vert(Y)s u c ht h a t{x (0) ,...,x (k) } span a k-simplex of X if and only if { (x (0) ),..., (x (k) )} span a k-simplex of Y.I fs u c ha is a bijection, the induced map on complexes ' : X ! Y is said to be a simplicial isomorphism [START_REF] Edelsbrunner | Computational topology, an introduction[END_REF]; equivalently, we say that X and Y are isomorphic.

We adopt simplicial isomorphism to build new sets of small k-simplices. In particular, we aim to determine convenient simplicial isomorphisms ' to obtain new sets of simplices that give unisolvent and minimal weights for P r Λ k (T ). The idea consists in starting from the set of nodes vert(X)=X 0 r (T )=L r (T ). Upon this set we may build the complex X = X r (T ). We consider a map that pushes points of L r (T ) to another set of nodes N r .W et h e nb u i l dt h ec o m p l e xY by asking to induce a simplicial isomorphism '.I no t h e rw o r d s ,{ (x (0) ),..., (x (k) )} is a k-simplex of the complex Y if and only if {x (0) ,...,x (k) } is a k-simplex of the complex X by construction.

x 0 x 1 x 2 x 3 x 0 x 1 x 2 x 3
The next step consists in restricting the choice on the above mentioned set of nodes N r . We shall consider only points that are unisolvent for P r (T )a n d well suited for nodal interpolation. There is a wide literature that treats this problem. Typically, Fekete nodes are invoked for obtaining optimality in the nodal context [START_REF] Roth | Nodal configurations and Voronoi Tessellations for triangular spectral elements[END_REF]. The main drawback with this approach is that they are hard to identify and hence to implement, since there is not an explicit formula. A trustful approximation is offered by warp and blend nodes [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF]. They come along with an explicit formula which is easy to implement. Also, they depend on a parameter ↵ that, if wisely chosen, gives an approximation of symmetrised Lobatto nodes studied in [START_REF] Blyth | A comparison of interpolation grids over the triangle or the tetrahedron[END_REF] as well. For these reasons in what follows we consider this last family. We will refer to symmetrised Lobatto nodes when ↵ =0 ,a st h e yi nf a c t mimic symmetrised Lobatto nodes, and to warp and blend nodes when ↵ = ↵ opt is optimised. Optimised values for ↵ can be found in [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF] and [START_REF] Hesthaven | Nodal discontinuous Galerkin methods[END_REF].

Upon these nodes we construct new family of k-simplices. Once the vertex map is given and the corresponding action on X r is a simplicial isomorphism ', we consider the image of set of small k-simplices X k r,min (T )={s 1 ,...,s N } under the action of '.W ep u t

Y k r,min (T ) . = N [ i=1 '(s i ). (3.35)
We write, for short, Y k r,min (T )=' X k r,min (T ) . It is not granted, a priori, that the so obtained set Y k r,min (T )i su n i s o l v e n tf o r P r Λ k (T ). In fact, in general, hypotheses of Theorem 2.28 are no longer satisfied. This problem is related with that of deducing unisolvence for points in R n ,w h i c h has been widely studied and whose solution is partial (see, for instance, [START_REF] Chung | On lattices admitting unique Lagrange interpolations[END_REF] and [START_REF]On certain configuration of points in R n which are unisolvent for polynomial interpolation[END_REF]). For this reason Proposition 3.2 comes to the rescue, allowing to check if weights associated with a so obtained set of simplices are unisolvent. Of course, if as e ti su n i s o l v e n t ,i ti sa l s om i n i m a lb yc o n s t r u c t i o n .

In the next section we present an explicit example of this construction based on warp and blend nodes. It is worth pointing out that optimality results are out of the scope and will not be claimed: in particular, choosing nodes that are optimised for nodal interpolation as vertices for simplices do not guarantee, a priori, that the corresponding simplices are optimal for the interpolation of differential k-forms as well.

Non uniform case: warp and blend simplices

We present a p ossible choice of simplicial isomorphism to generate different sets of simplices from X k r,min (T ). It is based on the mapping defined in [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF] (see also [START_REF] Hesthaven | Nodal discontinuous Galerkin methods[END_REF]) to obtain warp and blend nodes. The reason why we choose such a map is twofold. First, considering optimal nodes would not grant optimal simplices. Second, it offers an explicit implementation strategy. It also depends on a parameter and this allows easier comparisons with the existing literature, permitting the search for a relationship between nodal and simplicial results. We shall study this in detail in Chapter 4.

The warp and blend procedure consists of the composition of two different actions: the warping effect and the blending effect. Warping consists in defining vector fields w : F ! F that are tangent to (n 1)-faces F of the simplex T and redistributes the nodes on F .T h i se ff e c ti ss o f t e n e da so n em o v e st o w a r d st h e opposite vertex by the blending effect. The construction is carried by induction. We here follow [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF] and show the pro cedure for n = 2. A complete description for n = 3 can be found in [START_REF] Hesthaven | Nodal discontinuous Galerkin methods[END_REF]. Again, by induction one may generalise this construction to any n>3. The main difference with the existing literature is that we work on the standard simplex in place of the equilateral one.

For what concerns edges, i.e. when n =1 ,c o n s i d e rt h eG a u s s -L o b a t t o -Legendre nodes 1=p 0 <p 1 <...<p r =1o n [ 1,1]. They are optimal in this framework [START_REF] Fejér | Lagrangesche interpolation und die zugehörigen Konjugierten Punkte[END_REF]. We define a function that pushes the equidistributed points Chapter 3 1=x e 0 <x e 1 <...<x er = 1 to the above mentioned points as

w :[ 1, 1] ! [ 1, 1] w(x) . = r X i=0 (p i x e i ) j=r Y j=0 j6 =i ✓ x x e j x e i x e j ◆ . (3.36)
The effect of this map is depicted in Figure 3.5.

w(x) w(x) In order to approach the case n =2,(3.36)istransferredtoeac hedgeE of the 2-simplex F . Assume F =[ x 0 , x 1 , x 2 ]a n de n u m e r a t ei t se d g e sa sE 0 =[

x 1 , x 2 ], E 1 =[ x 0 , x 2 ], E 2 =[ x 0 , x 1 ]
. The only non vanishing barycentric coordinates on each edge E j are those associated with its vertices. So the map (3.36) applied to any edge E may be read with respect to these coordinates and we vectorise it by multiplying such a w for t E j ,theunitv ectortangen ttoE j .T h i sg i v e st h r e ev e c t o r fields defined on edges E 0 , E 1 , E 2 that extend to the whole F .I np a r t i c u l a r ,f o r j =0, 1, 2, we put

w j : F ! F w 0 ( 0 , 1 , 2 ) . = w( 2 1 )t E 0 w 1 ( 0 , 1 , 2 ) . = w( 0 2 )t E 1 w 2 ( 0 , 1 , 2 ) . = w( 1 0 )t E 2 .
Any w j is constant with respect to the barycentric coordinate j .L o o s e l ys p e a king, this means that the action of w j is independent on the distance between the point and the edge E j . This dependence is enforced by the blending effect. The blending function associated with the j-th edge is

b j : F ! R b j ( 0 , 1 , 2 ) . = 2 Y k=0 k6 =j 2 k 2 k + j . (3.37)
Notice that b j =1o nE j and b j (x j )=0 ,s i n c e j (x k )= j,k and P 2 k=0 k =1 , so the effect of b j is stronger near E j and weakens as one moves towards x j .

The desired warping and blending vector function is then obtained by summing the contributes along each edge and introducing an appropriate coefficient ↵ that draws nodes near or far from the edges:

g : F ! F g( 0 , 1 , 2 ) . = 2 X j=0 1+(↵ j ) 2 b j w j . (3.38)
Notice that symmetry is enforced as the same coefficient ↵ 2 R appears in front of each b i w i .T h ed e t e r m i n a t i o no f↵ is empirical and we refer to literature [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF].

The vector field (3.38) for ↵ = 0 is depicted in Figure 3.6. As stressed before, by induction one extends this construction to n 2. The corresponding formulation looks like that of (3.38) but counts n +1 summands. In the standard tetrahedron T the effect of warping and blending on the complex X 3 (T ) defined in (3.34) is depicted in Figure 3.7.

Unisolvence: computational evidences

A computational approach is thus necessary to check unisolvence of these sets. To this end, we use the generalised Vandermonde matrix and compute its determinant. We hence study its conditioning. We consider the k-simplices obtained from the uniform distribution by applying the simplicial isomorphism (3.38) discussed in the above section. We focus on two cases for the parameter ↵: ↵ =0 ,f o rw h i c h we obtain the so called symmetrised Lobatto [START_REF] Blyth | A Lobatto interpolation grid over the triangle[END_REF] simplices and an optimisation Chapter 4

From theory to practice

In Chapter 2 we studied the unisolvence of uniform small simplices as those introduced in [START_REF]High order edge elements on simplicial meshes[END_REF]. In Section 3.3 we offered a construction that allows to consider other families of small simplices, hence of weights, in a very flexible way; this was done by means of simplicial isomorphisms. Technical obstructions to prove their unisolvence have been addressed and solved by the computational approach introduced in Chapter 3. We have there introduced the generalised Lebesgue constant as a way for comparing different weights. This fact is made here precise and, to this end, interpolation onto P r Λ k (T )i si n v o l v e d . W ee n l i g h t e nt h er o l eo ft h e generalised Lebesgue constant for what concerns stability and local interpolation error, extending well known features of the nodal case.

After having discussed the interpolation operator induced by weights, we present several tables and compare generalised Lebesgue constants associated with different families of simplices. We focus both on the two and three dimensional framework. Results obtained for k =0a r ec o h e r e n tw i t ht h o s ea l r e a d yk n o w n in literature (see, for instance, [START_REF] Blyth | A comparison of interpolation grids over the triangle or the tetrahedron[END_REF], [START_REF] Blyth | A Lobatto interpolation grid over the triangle[END_REF] or [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF]) and we here present for the first time results associated with the case k = n.I n t e r e s t i n g l y ,i ti sp o s s i b l et oo b s e r v e as i m i l a rb e h a v i o u rb e t w e e nr e s u l t so b t a i n e df o rk-a n d( n k)-forms. This is evidenced by several plots.

Wide space is devoted to peculiarities of this setting. In particular, we verify the dependence of the generalised Lebesgue constant on the domain. This is carried by means of two different numerical experiments, which we also use to identify a sharp estimate. We close this chapter by suggesting a generalisation of the Runge counterexample, which will be theoretically motivated by the features of the interpolator.

Codes used to obtain symmetrised Lobatto and warp and blend nodes can be found in literature (see, in particular, [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF] and [START_REF] Hesthaven | Nodal discontinuous Galerkin methods[END_REF]). Their extensions to the construction of simplices, generalised Vandermonde matrices and generalised Lebesgue constant have been presented in [4].

The interpolation operator

Suppose {s 1 ,...,s N } is a unisolvent and minimal family of small simplices. The associated generalised Vandermonde matrix is a square, invertible matrix V ,a s shown in Proposition 3.2. This means that there is a unique element of P r Λ k (T ) whose weights on {s 1 ,...,s N } are N prescribed values. This allows to define the interpolation operator

Π k r :Λ k (T ) !P r Λ k (T )( 4 . 1 ) ! 7 ! Π k r ! such that Z s i ! = Z s i Π k r ! 8i =1,...,N. (4.2)
This definition is well posed. Indeed, assume by contradiction that there exist two interpolated Π k r ! and e Π k r ! for the same ! 2 Λ k (T ). Then Z

s i ! = Z s i Π k r ! = Z s i e Π k r ! 8i =1,...,N, whence Z s i ⇣ Π k r ! e Π k r ! ⌘ =0 8i =1,...,N.
By construction,

⇣ Π k r ! e Π k r ! ⌘ 2P r Λ k (T )
, and since {s 1 ,...,s N } is a unisolvent family, we have Π k r ! e Π k r ! =0,whenceΠ k r ! = e Π k r !. In terms of linear algebra, interpolating using (4.2) means fixing any basis {! 1 ,...,! N } for P r Λ k (T ), so that Π k r ! = a 1 ! 1 + ... + a N ! N ,a n ds o l v i n gt h e linear system 0

B @ R s 1 ! . . . R s N ! 1 C A = V 0 B @ a 1 . . . a N 1 C A , (4.3)
being V the generalised Vandermonde matrix written with respect to the basis {! 1 ,...,! N }. Hence, if {! s 1 ,...,! s N } is the cardinal basis associated with {s 1 ,...,s N },t h ei n t e r po l a t i o no pe r a t o r( 4 . 1 )bo i l sd o w nt o

Π k r ! = N X i=1 ✓Z s i ! ◆ ! s i , (4.4)
since V = I. Notice that the construction of the cardinal basis requires the computation of the inverse of V in any case, see Chapter 3.

Remark 4.1. Since S k 1 (T )=∆ k (T ), when r =1the interpolator (4.4) and that defined in (1.32) coincide.

We now show a p eculiar result that relates the op erators Π 0 r and Π 1 r . Proposition 4.2. Consider a unisolvent and minimal set S 1 r (T )={s 1 ,...,s N } of 1-simplices such that s i =[x i(0) , x i(1) ] and the associated interpolation operators Π 1 r and Π 0 r . Then the diagram

Λ 0 (T ) Λ 1 (T ) P r Λ 0 (T ) P r Λ 1 (T ) Π 0 r d Π 1 r d commutes.
Proof. We want to show that d (Π

0 r !)=Π 1 r (d!)f o re a c h! 2 Λ 0 (T ). Note that, since dΠ 0 r ! 2P r Λ 1 (T ), we have dΠ 0 r ! =Π 1 r (dΠ 0 r !), hence d Π 0 r ! = N X i=1 ✓Z s i dΠ 0 r ! ◆ ! s i = N X i=1 Π 0 r !(x i(1) ) Π 0 r !(x i(0) ) ! s i = N X i=1 !(x i(1) ) !(x i(0) ) ! s i = N X i=1 ✓Z s i d! ◆ ! s i =Π 1 r (d!),
where the second and the fourth equalities follow from Stokes' Theorem, while the first, the third and the fifth follow from the definition of interpolators (4.4).

Note that Proposition 4.2 cannot be extended to k>1, since there is not an analogous of Lemma 2.9 for spaces X k r,min (T )a n dt h u sS t o k e s 'T h e o r e mc a n n o t be applied. However, for k =1 ,t h i sr e s u l tr e l a t e st h em a t r i xo ft h eb o u n d a r y operator @ with that of the exterior derivative d acting on P r Λ 0 (T ), giving a pairing between small 1-simplices and Whitney 1-forms of degree r that reads as hd!, si = h!, @si . This was pointed out in [START_REF]High order edge elements on simplicial meshes[END_REF].

The interpolator (4.4) depends on the choice of weights, hence on supports of small simplices. Thus there is no reason for concluding that the best possible approximation of ! in P r Λ k (T ), i.e. the element of P r Λ k (T )t h a tm i n i m i s e s k! Π k r !k in some appropriate norm, is obtained for {s 1 ,...,s N } = X k r,min (T ). What can be claimed is that there shall be a (not unique in general) family of weights that gives

Π k r ! ⇤ 2P r Λ k (T )s u c ht h a t ||! Π k r ! ⇤ || 0 ||! Π k r !|| 0 8 Π k r ! 2P r Λ k (T ).
This holds in an appropriate norm, which is the 0-norm, introduced in Section 1.2. We call such a Π k r ! ⇤ the best fit approximation for !.I te n l i g h t e n st h er o l eo ft h e Lebesgue constant in interpolation of differential k-forms, since

||! Π k r !|| 0  (1 + Λ r ) ||! Π k r ! ⇤ || 0 . (4.5)
This was proved in [5, Proposition 2] and extends (3.17) to this context. Interpolation of differential k-forms onto P r Λ k (T )alsopro videsaw a ytoanalyse the quality of different families of weights. The bridge is again the generalised Lebesgue constant Λ r introduced in Chapter 3, which relates the stability of the (local) interpolation with the family of small simplices. The following result is in fact the simplicial counterpart of (3.16). It first appeared in [3].

||Π k r ! Π k r e !|| 0  "Λ r . (4.6)
Proof. By direct computation. Let S k r (T )={s 1 ,...,s N } be a family of unisolvent and minimal small simplices and {! s 1 ,...,! s N } the corresponding cardinal basis. Applying the linearity of Π k r and untangling its definition, we have

||Π k r ! Π k r e !|| 0 =s u p c2C k (T ) 1 |c| 0 Z c Π k r ! Π k r e ! =s u p c2C k (T ) 1 |c| 0 Z c Π k r (! e !) =s u p c2C k (T ) 1 |c| 0 N X i=1 Z c ✓Z s i (! e !) ◆ ! s i =s u p c2C k (T ) 1 |c| 0 N X i=1 Z s i (! e !) Z c ! s i .
The result now follows from the fact that S k r (T ) ⇢C k (T ) and the triangular inequality:

||Π k r ! Π k r e !|| 0  sup c2C k (T ) 1 |c| 0 N X i=1 Z s i (! e !) Z c ! s i =s u p c2C k (T ) 1 |c| 0 N X i=1 |s i | 0 |s i | 0 Z s i (! e !) Z c ! s i  " sup c2C k (T ) 1 |c| 0 N X i=1 |s i | 0 Z c ! s i = "Λ r ,
where the last inequality is a consequence of the fact that ||! e !|| 0  ".

The computation of this approximation of Λ r can be made efficient. Consider an ordering for the small simplices S k r (T )={s 1 ,...,s N } and one for a (finite) collection of k-simplices e C k (T ) . = {c 1 ,...,c M } that we will use in place of C k (T ). We offer a matrix formulation for the computation of the generalised Lebesgue constant. Choose any basis ω = {! 1 ,...,! N }.W eh a v es e e ni n( 3 . 8 )t h a tw ec a n compute the cardinal basis ω s = {! s 1 ,...,! s N } associated with S k r (T )b ys e t t i n g

V i,j = Z s i ! j , so that ω s = V T ω,
thinking of ω and ω s as column vectors. Now, we compute for any k =1,...,M, Z

c k ! s i = Z c k N X j=1 V T i,j ! j = N X j=1 V T i,j Z c k ! j (4.7)
and thus put the M ⇥ N matrix W whose (k, j)-th entry is

W k,j . = Z c k ! j . (4.8)
Now, plugging (4.8) into (4.7), we obtain the auxiliary matrix

L . = WV T , (4.9)
such that

L i,j = Z c i ! s j .
Define the matrices S 2M N,N and C 2M M,M such that S i,j . = ( |s i | 0 if i = j 0o t h e r w i s e and C i,j . =

( |c i | 1 0 if i = j 0o t h e r w i s e . We define L . = CWV T S, (4.10)
to obtain that Λ r = kLk 1 , being k•k 1 the 1-norm for matrices.

We have provided an exact way to compute the matrices V and W by using Equation (3.9). Since it is extremely expensive and unpractical, in this context quadrature rules that exactly integrate polynomials up to the desired degree are involved. We address to [4] for detailed algorithms and codes.

Lebesgue constants in R 2

Computations of nodal Lebesgue constants on a triangle have been carried for several set of nodes, see, for instance, [START_REF] Roth | Nodal configurations and Voronoi Tessellations for triangular spectral elements[END_REF], [START_REF] Pasquetti | Spectral element methods on unstructured meshes: which interpolation points?[END_REF] and [START_REF] Luo | A Lobatto interpolation grid in the tetrahedron[END_REF]. We here show results for k =1andk =2,thatis,resultsassociatedwith1-and2-simplices. W efirstrecall results for k =0 ,t h a tw er e c o m p u t e do nas a m p l es e to f5 2 8po i n t sc o n t a i n e di n the standard triangle T .T h e ya r eg a t h e r e di nT a b l e4 . 1 .

In accordance with Section 3.2.1, we stress that results for k = 0 may be compared with known literature. Small differences may be due to the fact that the reference cloud of points on which the supremum is checked might be different. For what concerns k =1 ,r e s u l t sh a v eb e e nfi r s tc o m p u t e di n [ 3 ] . I ns u c hac a s e the standard triangle T =[0, e 1 , e 2 ]w a sc o n s i d e r e da sw e l l .

As we shall deduce from tables below, non uniform distribution of simplices (i.e. distribution of simplices whose vertices are not uniformly distributed in T )willoffer lower Lebesgue constants. However, Remark 4.4 will clarify the necessity of wisely choosing non uniform simplices. In particular, it will show that a complete lack of (rotational) symmetry will even worsen performances with respect to uniform simplices. Throughout the chapter we thus focus on symmetrised Lobatto and warp and blend simplices, whose construction has been detailed in Section 3.4.

An interesting comparison is carried in Figure 4.4, which seems to suggest an intimate relationship between the nodal behaviour and that of simplices based on the corresponding set of points.

k=1

We b egin with the case of k = 1. We consider a mesh generated by the software Triangle that presents 513 edges with length between 0.011 and 0.120, and compare Lebesgue constants of X 1 r,min (T ), which we refer to as uniform simplices, symmetrised Lobatto and warp and blend simplices in Table 4 Case k =1 : L e b e s g u ec o n s t a n t si nat r i a n g l eT ⇢ R 2 ,a s s o c i a t e dw i t hu n i f o r m (X 1 r,min (T )) and nonuniform (constructed using the simplicial isomorphism given in Equation (3.38) for ↵ =0and↵ = ↵ opt )distributionsofsmalledgesfordifferent polynomial degrees r 1. Results are reported in Table 4.2 on the left. A visual comparison in semi-logarithmic scale is offered in Figure 4.1 on the right. Results associated with uniform simplices are depicted in red, whereas blue and green are associated with non uniform weights.

In accordance with well known results for the nodal case, we see that non uniform distribution of simplices offer lower generalised Lebesgue constants. Such non uniform distributions are obtained by enforcing a significant rotational symmetry on the simplices. In Remark 4.4 we also consider a case in which we weaken this request. Performances are discouraging and we will thus abandon this road.

Remark 4.4. Notice that we have considered only uniform, symmetrised Lobatto and warp and blend simplices. We made this choice observing that configurations that do not respect symmetry yield poor results. As an example we show the following and then drop this kind of situations.

Consider a triangle T ⇢ R 2 and Gauss-Lobatto-Legendre nodes on each edge. Choose two edges E 1 and E 2 and join them with E 0 by drawing lines that are parallel to E 2 and E 1 , respectively, starting from Gauss-Lobatto-Legendre nodes on E 1 and E 2 . This situation is depicted in Figure 4.2 for r =3. This is obtained by a simplicial isomorphism from X 2 r,min (T ).

x

k=2

The computation of these constants is a novelty. To obtain it, we consider a family of 300 simplices supported in T and reproduce the computation done before in the case k = 1. Unisolvence and minimality of the considered sets of small simplices are a consequence of Table 3 Results for k =2confirmthetrendsobserv edfork =0andk =1,underlying a better behaviour for weights associated with non uniform simplices.

Comparisons for 0-, 1-and 2-simplices based on the same set of nodes are provided in Figure 4.4. We essentially see a parallelism of values of Lebesgue constants for k =0 ,k =1a n dk =2a s s oc i a t e dw i t hk-simplices associated with the same set of nodes. The bending that one may observe for 2-simplices in the cases of symmetrised Lobatto and warp and blend nodes is to be imputed to the poverty of the mesh.

Lebesgue constants in R 3

Computation of nodal Lebesgue constants on a tetrahedron have been studied in several works and for a large choice of points. In particular, interesting computations for nodes that we here consider as vertices for the small simplices, such as symmetrised Lobatto and warp and blend, are carried in [START_REF] Luo | A Lobatto interpolation grid in the tetrahedron[END_REF], [START_REF] Blyth | A comparison of interpolation grids over the triangle or the tetrahedron[END_REF] and [START_REF] Hesthaven | Nodal discontinuous Galerkin methods[END_REF]. We first recall such results for k =0andgathertheminT able4.5.

We repro duce the same analysis done for a triangle T ⇢ R 2 .W ec o n s i d e rt h e standard tetrahedron T =[0, e 1 , e 2 , e 3 ]. Again, results for k = 0 may be compared with those already known in the above mentioned literature in view of Proposition 3. [START_REF] Blyth | A Lobatto interpolation grid over the triangle[END_REF]. For what concerns k =1 ,t h e s ec o m p u t a t i o n sh a v eb e e np e r f o r m e da n d presented in [3], where a comparison with a different tetrahedron have been offered as well. This is reproposed here in Section 4.2.1. For what concerns k =2,results may be also found in [2]. In Table 4.9 we present a comparison when the simplex T is varied, checking estimates of Section 3.2.1. Results for k =3a r ean o v e l t y and agree with the known ones for k =0,k =1andk =2sho wingthatabetter behaviour is associated with nonuniform simplices. k=1

For what concerns k = 1, we consider a reference set of 13800 edges whose lengths vary between 0.0435 and 0.0615. In accordance with the nodal case, Lebesgue constants associated with uniform simplices grow faster than those associated with non uniform ones. This is the content of Table 4.6, which is plotted in Figure 4.5.

uniform nonuniform r X For what concerns k = 2, we consider a reference set of 3876 faces supported on the standard 3-simplex T . Areas of the faces range between 0.0017 and 0.0030. Estimated Lebesgue constants over such a set are reported in Table 4.7. It is worth noting that both their trend and their magnitude recall those obtained for k =1, that are reported in Table 4 Elements of the sets here studied have been depicted in Figure 3.4. As for other cases presented, we see that non uniform weights offer smaller Lebesgue constants. Table 4.7 first app eared in [2].

k=3

For what concerns k =3,w econsiderareferencemeshof2300tetrahedra. These tetrahedra are uniformly distributed in T ,h e n c et h e ym a ys l i g h t l ys l o wd o w n the growth of uniform small simplices. Nevertheless, such a family shows the worst results of Table 4.8. In the comparison of Figure 4.8, this can be read by observing that the curve associated with results for k =3c r o s s e st h a tf o rk =0 . A visual depiction of the sets here studied is given in Figure 3 In general, we see that weights associated with non uniform simplices offer lower Lebesgue constants. This is in accordance with the well known literature for the nodal case. Therefore, Proposition 4.3 and Equation (4.5) suggest that the nonuniform choice is safer for interpolation of differential k-forms. We shall in fact see in Section 4.4 that Runge-like phenomena arise in this context as well when dealing with large Lebesgue constants.

Comparisons in semi-logarithmic scale for 0-, 1-, 2-and 3-simplices based on the same set of nodes are provided in Figure 4.8.

Results are in accordance with those shown in Figure 4.4 concerning the two dimensional case. The behaviour of a family of k-simplices seems again to inherit its trend from the set of nodes on which it is constructed. Also, there is a relevant closeness between results for k-and(n k)-simplices (that is, for 0-and 3-simplices and 1-and 2-simplices). This can be explained in terms of the Hodge operator, which relates k-a n d( n k)-forms. Interestingly, another result which relates 0and n-forms, in this context, was already given in Proposition 3.13 and Proposition 3.14.

We dedicate the next section to observing how these results change when the simplex T is changed.

Changing the simplex

We dedicated Section 3.2.1 to studying the relationship that o ccurs b etween the generalised Lebesgue constant computed on a simplex T and that computed on another simplex T ' obtained as the image of T under a non degenerate affinity '(x)=M x + b.T h i sp r o b l e mh a sa l s ob e e nt r e a t e di n [ 3 ] .R e c a l lt h a tw ep r o v e d Λ r (T ' )  k 2 (M )Λ r (T ), being 2 (M )t h e2 -c o n d i t i o n i n go ft h em a t r i xM .W et h u sp l a c eo u r s e l v e si nt h e three dimensional space R 3 and make two different experiments. In this section we stress the dependence of the Lebesgue constant on the simplex and thus specify it.

A first experiment

To b egin with, we consider two fixed tetrahedra and let the p olynomial degree r vary. We explicitly compute the affinity ' and verify the above theoretical bound.

As simplex T we consider the standard 3-simplex, whereas for T ' we choose the equilateral tetrahedron proposed in [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF]. It has side lengths equal to 2 and it is centred at the origin. Hence it is generated by vertices

v 0 = ⇣ 1, 1 p 3 , 1 p 6 ⌘ , v 1 = ⇣ 1, 1 p 3 , 1 p 6 ⌘ , v 2 = ⇣ 0, 2 p 3 , 1 p 6 ⌘ , v 3 = ⇣ 0, 0, 3 p 6

⌘

.T h ea ffi n i t y' thus reads as Table 4.9: For k =1: ratios Λr(Tϕ) Λr(T ) of Lebesgue constants in tetrahedra T ' and T for edges associated with uniform, symmetrised Lobatto and warp and blend nodes. For k =2 : r a t i o s Λr(Tϕ) Λr(T ) of Lebesgue constants in tetrahedra T ' and T for faces associated with uniform, symmetrised Lobatto and warp and blend nodes. Here r is the total polynomial degree and T ' is generated by vertices

'(x)= 0 @ 21 1 0 3 p 3 1 p 3 00 4 p 6 1 A x + 0 @ 1 1 p 3
v 0 = ⇣ 1, 1 p 3 , 1 p 6 ⌘ , v 1 = ⇣ 1, 1 p 3 , 1 p 6 ⌘ , v 2 = ⇣ 0, 2 p 3 , 1 p 6 ⌘ , v 3 = ⇣ 0, 0, 3 p 6 ⌘ .
To conclude this first experiment, we show an interesting fact. We claimed the centrality of the hypothesis of M being diagonal in Proposition 3.16, asserting that such a results cannot be extended even to the case of diagonalisable matrices, as one could guess, since changing the basis of R n to make M diagonal would mean changing coordinates of vertices of T but not its shape. Now, the matrix M associated with ' is diagonalisable, since it has three different real eigenvalues. In

A second experiment

For what concerns the second exp eriment we do the following. We fix the p olynomial degree and shrink the standard tetrahedron T by means of a family of affinities depending on a parameter .I n p a r t i c u l a r , w e k e e p fi x e d t h e v e r t e x v 0 =(0, 0, 0) and the height of T equal to 1 p 3 and move vertices v 1 , v 2 and v 3 by the affinity ' (x)=M x given by

' (x)= 0 @ 1 2 1 2 1 2 1 A x,
to obtain different simplices T . = ' (T ). One has 2 (M )= 1 1 3 .W et h u sc o n s i d e r r =7a n dl e t vary. We compare Λr(T δ ) Λr(T ) with 2 (M )a s tends to

vertices v 0 =( 0 , 0, 0), v 1 =( 1 2 , , ), v 2 =( , 1 2 , ), v 3 =( , , 1
k r ! Π k r e !k 0  Λ r , (4.11)
provided that k! e !k 0  ".I n S e c t i o n 4 . 2 w e h a v e e s t i m a t e d t h e r i g h t h a n d side of the above inequality, and we now offer numerical experiment that compute the left hand side, verifying numerically Proposition 4.3. Some examples of these computations are contained in [3] for what concerns edge elements an in [2] for the case of face elements. Now, pick any basis {! 1 ,...,! N } of P r Λ k (T )a n dau n i s o l v e n ta n dm i n i m a l set S k r (T ). For any ! 2 Λ k (T ), Π k r ! is computed as in (4.2), and Π k r e ! is obtained in the same way but starting from perturbed weights R s i ! + " i .I no r d e rt oc o n t r o l such a perturbation " i and to be consistent with the request k! e !k 0 ,w econsider ar a n d o mpe r t u r b a t i o n✏ i 2 [0, 1] and put

1 |s i | Z s i (! e !) .
=(2✏ i 1)" (4.12)

for i =1 ,...,N and we let " vary from 10 1 to 10 8 .T h ek-form e ! is obtained consequently. This yields

Π k r ! = N X i=1 ✓Z s i ! ◆ ! s i and Π k r e ! = N X i=1 ✓Z s i ! + " i ◆ ! s i .
In order to measure kΠ k r ! Π k r e !k 0 and make such quantity consistent with Λ r , we consider the same collection of k-simplices e C k (T ) . = {c 1 ,...,c M } that we used to estimate Λ r .T h i sc o m p u t a t i o ni sc a r r i e di nMatlab,w h e r et h er i g h th a n ds i d e of (4.12) reads (2 ⇤ rand(1) 1) ⇤ ". We thus produce numerical experiments that compare the quantities 1 " ||Π k r ! Π k r e !|| 0 for " =1 0 2 , 10 5 , 10 8 and match them with the Lebesgue constants offered in previous Subsections. Noting that this quantity depend, in fact, only on (4.12), we get the following algorithm.

A =(W ⇤ (V \ ))./" compute zeronorm(r,`)=norm(A, 0 inf 0 ) end end
Results for uniform, symmetrised Lobatto and warp and blend nodes for n =2 and n = 3 are depicted below. As an example, we present only one table, which is Table 4.14 and refers to k =1a n dn =2 ,s t r e s s i n gt h a ti ng e n e r a lt h e ya r e not reproducible for the rand term. Also, to make the work more readable, since results for symmetrised Lobatto and warp and blend simplices are very close, we present only graphs for the case of ↵ = ↵ opt .W ea l s oi n v i t et h er e a d e rt om a t c h these results with those contained in [3] for k =1and [2]fork =2.

A generalisation of the Runge phenomenon

The importance of reducing the generalised Lebesgue constant when dealing with (local) interpolation is underlined by Equation (4.5). A classical explanation, in one dimension, is provided by the Runge phenomenon,w h i c hs h o w st h a ti n t e r p olation based on equidistributed nodes yields poor approximation properties [START_REF] Runge | Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten[END_REF]. In particular, Runge showed that there exists a smooth function f :[ 1, 1] ! R such that the norm kf Π 0 r f k 0 (that is, the sup-norm for functions) grows as the polynomial degree r of the interpolated Π 0 r f grows. Exploiting several variables Taylor expansions [START_REF] Ciarlet | General Lagrange and Hermite interpolation in R n with applications to finite element methods[END_REF], a higher dimensional counterpart of this phenomenon can be theorised. Numerical generalisations to functions R n ! R and points that blend this effect have been studied, mostly for n = 2 and n =3 ,b yc o n s i d e r i n g Fekete p oints [START_REF] Roth | Nodal configurations and Voronoi Tessellations for triangular spectral elements[END_REF] or appropriate approximation of such no des. We use Prop osition 4.2 to show that the same phenomenon shows up also in this context by taking ! =df 2 Λ 1 (T ), being T the standard n-simplex and

f : T ! R f (x 1 ,...,x n )= n Y i=1 1 1+100 x i 1 2 2 .
(4.13) the generalised Runge function [START_REF] Roth | Nodal configurations and Voronoi Tessellations for triangular spectral elements[END_REF]. The behaviour of this function for n =2and n =3a n dt h ei n t e r po l a t i o npe r f o r m a n c e so fs e v e r a ln od e sh a v ebe e na n a l y s e di n many works, such as [START_REF] Blyth | A comparison of interpolation grids over the triangle or the tetrahedron[END_REF], [START_REF] Blyth | A Lobatto interpolation grid over the triangle[END_REF] and [START_REF] Luo | A Lobatto interpolation grid in the tetrahedron[END_REF]. A couple of remarks are needed: the term x i 1 2 centres the function outside the interior of the simplex for n>1. In other contexts this is substituted by x i 1 n to centre the function f on the barycentre of the standard simplex T .R e s u l t sa r ec o m p a r a b l e ,a n dw ea d o p tt h i sc h o i c et o stick with [START_REF] Luo | A Lobatto interpolation grid in the tetrahedron[END_REF]. The coefficient 100 in front of this term is due to the translation to the standard simplex. This is better understood in one dimension, by mapping (by affinity) the classical Runge function from [ 1,1] to the standard 1-simplex in R,w h i c hb o i l sd o w nt ot h eu n i ti n t e r v a l[ 0 , 1]. Last, since we defined f on the standard simplex T ,E q u a t i o n( 4 . 1 3 )c a nbee a s i l yr e a di nb a r y c e n t r i cc oo r d i n a t e s by applying the change of coordinate x i = i .W ec a l lRunge 1-form the element of Λ 1 (T )o b t a i n e da st h ed i ff e r e n t i a lo f( 4 . 1 3 ) ,w h i c hi s

! . = n X i=1 @f @x i dx i .
We construct Π 1 r ! as in (4.4) for weights associated with different sets of small 1-simplices and measure the error k! Π 1 r !k 0 in the 0-norm as the degree r of the interpolated Π 1 r ! increases. Results for R 2 are reported in Table 4.15, whereas for R 3 are reported in Table 4.16. Trends of the quantities k! Π 1 r !k 0 associated with different weights with respect to r are compared in Figure 4.14 for R 2 and Figure 4.15 for R 3 . Meshes considered are the same as those used for computing the Lebesgue constants and the stability tests. The numerical scheme is the following. 4.15, respectively) show that uniform small simplices do not offer good interpolation properties. This is in accordance with the nodal case and verifies Equation (4.5) on the numerical level. However, in contrast with usual results for the Runge phenomenon on an interval, there seems not to be an optimal distribution of simplices that drastically reduces this generalisation of the issue. Of course, this is due to the fact that generalised Lebesgue constants associated with 1-simplices studied above improve results obtained for uniform simplices, but they are still far from those obtained for the nodal case, see and compare Table 4.5 with Table 4.6 on page 96.

Conclusions

In this thesis I have answered several questions: to begin with, I have identified unisolvent and minimal sets of the celebrated small simplices X k r (T ). This proof is completely new, and so far only a proof of unisolvence was known [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF].

A systematic construction of sets of uniform small simplices that induce minimal and unisolvent weights was in fact provided in Chapter 2. This was denoted by X k r,min (T ), and the rule proved does not require restriction on the dimensions k and n nor on the polynomial degree r.I np a r t i c u l a r ,Io ff e r e dar e c i p ef o rd i scarding redundant elements of X k r (T )thatisconsisten twiththeruleexploitedfor extracting a basis from a system of generators for P r Λ k (T )p r e s e n t e di nC h a p t e r 1. Proving unisolvence of this new set required the introduction of an auxiliary set with similar properties, Σ k r,min (T ). The difficulties in obtaining such results are wide: first, their generality makes it hard to handle quantities at play. Second, if one tries to reproduce low dimensional examples to fix ideas, then it is easy to fall into too strong simplifications. For instance, any case studied in R 2 or even R 3 can be deduced by peculiar cases. This has been deepened at the end of Chapter 2. In fact, when k =0o rk = n the majority of these results can be treated as usual interpolation issues, whereas when k =1o rk = n 1i ti sp o s s i b l et ob o i l down to specific simplification, as shown in Section 2.7. For this reason visualising these examples can be misleading. Third, and this is the most important aspect, such results are as sharp as possible: as a consequence one has to be very accurate in definition of mappings and checking all the specific cases. It is worth noting the difference between the proof of Lemma 2.14 (non sharp case) and that of Lemma 2.18 (sharp case), which play a parallel role.

I then developed, sharpened and studied numerical tools to treat the problem of unisolvence under different points of view. This was done in the first part of Chapter 3. The first instrument that was defined is the generalised Vandermonde matrix. Such a matrix allows to transform the problem of deducing unisolvence into that of checking the rank of the matrix associated with a bilinear form. This bilinear form, which is obtained by integration, represents the pairing of P r Λ k (T ) and a given set of small simplices. This construction was thus used to identify a preferred basis for the space of trimmed polynomial differential k-forms. It was called the cardinal basis and its construction is equivalent to making the generalised Vandermonde matrix equal to the identity. This procedure clarifies why trimmed polynomial differential forms are entitled to be the natural generalisation of Whitney forms. Since these results are carried in good generality and do not exploit the features of the set of small simplices, they are a valuable alternative when a direct proof of unisolvence is hard or even impossible to carry. This opens the way to the definition of generalised Lebesgue constant, a generalisation of the classical Lebesgue constant adopted in nodal interpolation. It allowes comparisons between different choices of simplices. The path towards nonuniformity has thus been traced and the problem of proving the corresponding features of unisolvence and minimality is then systematically transferred to checking the rank of the generalised Vandermonde matrix. Computational results show the necessity of nonuniform sets. In fact, numerical tests performed in Chapter 4d i s c r e d i tt h es e tΣ k r,min (T ) and its closest variants obtained by linearity of the integral (see Table 4.13) compared to X k r (T ). Also, results concerning small simplices X k r (T )s h o w e da ni m p o r t a n tg r o w t ho ft h ea s s o c i a t e dL e b e s g u ec o n s t a n t . This forced us to introduce different families of small simplices.

The most natural choice was then that of deforming existing simplices without changing the global shape of the domain. Since both the sets X k r (T )andX k r,min (T ) bear a structure of simplicial complex, a convenient strategy was that of keeping it and I thus looked for isomorphic complexes. A comparison with the well known nodal case suggested to consider as vertices of the k-simplices nodes that were suitable for nodal interpolation. Hence, on the one hand the necessity of explicitly compute them and on the other the requirement on the topology, lead to warping and blending mappings. In the second part of Chapter 3, specifically in Section 3.3, It h u so ff e r e dad e t a i l e dc o n s t r u c t i o nt h a tc a nbea p p l i e dt ot h ea bo v em e n t i o n e d sets of small simplices to obtain new ones. As expected, numerical tests carried in Chapter 4 not only did show the unisolvence of the so obtained weights, but also highlighted a significant improvement of their Lebesgue constant, which was proved to measure, in an appropriate sense, the quality of the weights.

Before turning to numerical results, I dealt with some features of the Lebesgue constant and an interesting behaviour was observed and pointed out. In fact, on the one hand it was shown that classical issues of nodal interpolation can be generalised and addressed also in the case of simplicial interpolation. In particular, Ip r o v e dt h a ts u c ha ni n d i c a t o ri nf a c tc o n t r o l st h es t a b i l i t yo fl oc a li n t e r po l a t i o n . On the other hand, however, it was proved that such a quantity, when k 6 =0 or k 6 = n,b e h a v e sd i ff e r e n t l yf r o mt h en o d a lo n e . I nf a c t ,i ft h eu s u a lL e b e s g u e constant is universal, in the sense that it reads only the relative positioning on points inside the domain and not the placement of the domain itself, the generalised one is seriously affected by the shape of the simplex T . I thus computed bounds that relate generalised Lebesgue constants computed on different simplices with the properties of the geometrical mapping that sends a simplex into the other. Numerical verifications of Chapter 4 are consistent with theoretical predictions.

Numerical tests of Chapter 4 also confirmed the theoretical results of Chapter 2a n dS e c t i o n3 . 3 . I np a r t i c u l a r ,a se x p e c t e d ,i tw a se n l i g h t e n e dt h a tn o n u n i f o r m supports improve both the stability and the quality of the interpolation. This was mainly observed by means of the generalised Lebesgue constant. I also provided ap a t h o l o g i cs i t u a t i o nt h a ts h o w sag e n e r a l i s a t i o no ft h eR u n g ep h e n o m e n o nt o the case of differential forms. This has its roots in Stokes' Theorem and numerical experiments clearly highlighted the issue.

To summarise the results obtained, I may conclude that weights are a compelling alternative to moments. The main drawback of this approach consists in the lack of a systematic treatment, so proving results about specific sets of small simplices can be long and technical. An evident example is that in order to prove unisolvence of the set X k r,min (T )i tw a sn e c e s s a r yt oi n t r o d u c ea n ds t u d yt h es e t Σ k r,min (T ). Despite this, if one is interested in proving unisolvence for a specific set, and if the polynomial degree r is fixed, it is easy to construct the generalised Vandermonde matrix and check its rank to get unisolvence. This fact yields an incredible flexibility and widely enriches the potential sets of small simplices. In particular, this makes it possible to put at play nonuniform simplices, whose performances improve the conditioning of the generalised Vandermonde matrix and sharply reduce the associated generalised Lebesgue constant, hence yielding interesting stability and interpolation properties. As an example of interest of this, it is worth citing the capability of managing the Runge phenomenon as in the nodal case.

Iaminapositiontoaddressafewquestionsforfurtherin v estigationsonthese topics. First of all, a systematic proof of unisolvence for generic support of weights still lacks. Techniques of Section 3.1 offer some partial results in this perspective.

Acrucialaspectisthestudyofoptimal simplices and the behaviour of the corresponding weights. In this work only distribution of points inherited from nodal interpolation were considered as vertices of the support for the weights. Such nodes are obtained by imposing some requirements that are peculiar to nodal interpolation, for instance on the determinant of the associated (nodal) Vandermonde matrix. As pointed out in Chapter 3, there is no reason for considering the so obtained supports for weights as the best possible for simplicial interpolation as well, even though they offered convincing improvements with respect to uniform small simplices. Chapter 4 gave some insights in this direction, stressing the role of rotational symmetry in this context. With all this in mind, a search for optimal simplices may be examined in depth. A first step for further improving numerical results might be that of recomputing and optimising the parameters known in lit-erature for warp and blend nodes, looking for values that minimise the generalised Lebesgue constant for each k>0. As a more challenging problem, both on the theoretical and the computational side, one shall try to apply techniques for the search of optimal distribution of nodes to the case of simplices, for instance looking for simplices that maximise the determinant of the generalised Vandermonde matrix, and check if the corresponding weights sharpen results here presented.

To conclude, in order to strengthen the approach here presented, the most interesting and solid development would be that of adopting techniques presented in this work to solve a concrete physical problem modelled by a PDE. In literature one may find some applications to the Darcy problem or the model problem offered in [START_REF]High order edge elements on simplicial meshes[END_REF]. To begin with, I aim to treat these problems with nonuniform weights to quantify the improvement given by novelty introduced in this work. A subsequent application to electromagnetism is in my future plans.

1 . 1 4 From theory to practice 4 . 1

 11441 Simplices and complexes . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Differential forms on simplices . . . . . . . . . . . . . . . . . . . . . 1.3 High order Whitney forms . . . . . . . . . . . . . . . . . . . . . . . 2 Weights, unisolvence and minimality 2.1 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Small simplices X k r (T )......................... 2.3 Unisolvence of X k r (T ). . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The set Σ k r (T )a n di t su n i s o l v e n c e . . . . . . . . . . . . . . . . . . . 2.5 Unisolvence and minimality of Σ k r,min (T )............... 2.6 Unisolvence and minimality of X k r,min (T )............... 2.7 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Non uniform high order weights 3.1 The generalised Vandermonde matrix . . . . . . . . . . . . . . . . . 3.1.1 Exact computation of the generalised Vandermonde matrix . 3.2 The generalised Lebesgue constant . . . . . . . . . . . . . . . . . . 3.2.1 Dependence on the simplex . . . . . . . . . . . . . . . . . . 3.3 Simplicial isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Non uniform case: warp and blend simplices . . . . . . . . . . . . . 3.4.1 Unisolvence: computational evidences . . . . . . . . . . . . . The interpolation operator . . . . . . . . . . . . . . . . . . . . . . . 4.2 Computation of Lebesgue constants . . . . . . . . . . . . . . . . . . 4.2.1 Changing the simplex . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Non simplicial supports . . . . . . . . . . . . . . . . . . . . 4.3 Stability results: verification of the Lebesgue constants . . . . . . . 4.4 A generalisation of the Runge phenomenon . . . . . . . . . . . . . .

  iv

  sav e r t e xm a pa n di n d u c e s as i m p l i c i a lm a p ,b u ti td o e sn o tp r e s e r v ea n yt o p o l o g i c a ls t r u c t u r eo fT .W ea r e thus interested in a peculiar class of simplicial maps, which is motivated by the following Lemma. Lemma 1.5 ([50], Lemma 2.8). Let X and Y be two simplicial complexes. Let :v ert(X) ! vert(Y) be a bijection such that {x (0) ,...,x (k) } span a simplex of X if and only if { (x (0) ),..., (x (k) )} span a simplex of Y. Then the induced map ' : |X|!|Y| is a homeomorphism.

Figure 1 . 3 :

 13 Figure 1.3: Some examples of Whitney forms associated with their generating simplex. To check the equality ! T =dvol T =d 1 ^d 2 ^d 3 apply the definition of ! T and substitute d 0 = d 1 d 2 d 3 .

Figure 2 . 2 :

 22 Figure 2.2: A visual comparison of the sets L r (T ) (filled and empty circles) and Z r (T )( o n l yfi l l e dc i r c l e s ) ,f o rr =3. The cardinality of L r (T )is n+r r ,whenceonegetsthatZ r (T )consistsof n+r 1 r 1

Figure 2 . 4 :

 24 Figure 2.4: The midpoint m of the edge [x 0 , x 1 ]m a yb et h o u g ho ft h ei m a g eo f both vertices x 0 and x 1 .

  so b t a i n e df r o mak-face F 2 ∆ k (T )b ym e a n so f ξ . Hence p(ξ)=Π k T ( ξ !)i sap o l y n o m i a lo fd e g r e er 1i nξ which is zero by hypothesis for each ξ 2 Z r (T ); thus p(ξ)=0f o re a c hξ 2 R n .B yL e m m a2 . 1 5 ,w ec o n c l u d e that ! =0.

2. 4 0 Figure 2 . 5 :

 4025 Figure 2.5: The set Σ k r (T ), for r = 3. For graphical reasons, we chose x 0 =(0, 0, 1).

Figure 2 . 7 :

 27 Figure 2.7: Left: the set Σ 2 r,min (T ), for r =3 . R i g h t : t h er e d u n d a n ts i m p l i c e si n Σ 2 r (T ), for r =3,i.e. thecollectionΣ 2 r (T ) \ Σ 2 r,min (T ).

Figure 2 . 8 :

 28 Figure 2.8: A section of T ⇢ R 3 that shows the face F 0 =[x 1 , x 2 , x 3 ]. The dashed 2-simplices are redundant, but belong to Σ 2 3 (T ).

Figure 2 . 9 :

 29 Figure 2.9: The segments supported on [x 1 , x 2 ]o fΣ 1 3 (T )( l e f t )o v e r l a ps e v e r a l times; for instance, the middle segment belongs to four different elements of Σ 1 3 (T ) (orange, blue, cyan and black). The same segment in the picture on the right, which represents the set Σ 1 3,min (T )a n dh i g h l i g h t st h o s es u p p o r t e do n[ x 1 , x 2 ], belongs only to two elements of Σ 1 3,min (T )( c y a n ,b l a c k ) .

  T ). Then ! =0. Proof. For k = n and k =0t h e r ei sn o t h i n gt op r o v e . I nf a c t ,b yL e m m a 2.22 in those cases Σ k r,min (T )=Σ k r (T ), hence the result follows immediately from Proposition 2.20.

Figure 2 .

 2 Figure 2.11: A visual depiction of X 2 2,min (T )f o rT ⇢ R 3 .T h e d a s h e d t r i a n g l e is the only element of X 2 2 (T )t h a td o e sn o tb e l o n gt oX 2 2,min (T ), assuming that x 0 = 0.

. 21 )

 21 Since !| Fσ =0f o r 6 =i d ,t h e n!| ⇡σ,α 2P r 1 Λ k (⇡ ,α )f o rj =1 ,h e n c eb y Proposition 2.17 we conclude that !| ⇡σ,α =0f o ra l lα such that |α| = 1. As a consequence,R s 0 ! =0f o re a c hs 0 2 Σ k r 1,min (T ) \ ⇡ ,α .O n ec a nc o n t i n u ei nt h i s way for |α| =2,...,r 1 to deduce with the same argument that !| ⇡σ,α =0,hence R s 0 ! =0foreac hs 0 2 Σ k r,min (T ) \ ⇡ ,α ,f o r 6 =idand|α|2{1,...,r 1}.S i n c e the case =i di sp r o v i d e db y( 2 . 1 9 ) ,w eh a v e R s 0 ! =0f o re a c hs 0 2 Σ k r,min (T ), hence one has ! =0b yProposition2.25.

  (T )f o rat e t r a h e d r o nT ⇢ R 3 with respect to the total polynomial degree r.W e c o m p a r e t h e m o n o m i a l b a s i s λ α ! F for P r Λ k (T )( l e f t ,s e c o n dt o fourth column) with the Bernstein basis B α ! F (three rightmost columns).

  .[START_REF]On certain configuration of points in R n which are unisolvent for polynomial interpolation[END_REF] i.e. as the supremum of the (absolute values of the) Lebesgue function,w h i c hi s the sum of the elements of the cardinal basis associated with a collection of points {p 1 ,...,p N },b e i n gN =dimP r (I)a n dI ⇢ R.I ns u c hac a s ea ne x p l i c i tf o r m u l a for the cardinal basis is given (see [27, Equation (2.5.1)]) and is

9 Figure 3 . 2 :

 932 Figure 3.2: Equispaced nodes yield a high Lebesgue constant. In fact, as the polynomial degree increases, the interpolated constructed over these diverges from the Runge function f (x)= 1 1+25x 2 .T h i sw a sn o t e db yC a r lR u n g ei n1 9 0 1[ 6 2 ] .

Figure 3 . 4 :

 34 Figure 3.4: An example of simplicial isomorphism: blue shaded 2-simplices are mapped preserving adjacency and vertex enumeration.

Figure 3 . 5 :

 35 Figure 3.5: The effect of the mapping w in (3.36) on [ 1, 1] for r =5 . E q u i d i stributed points (black) are moved to Gauss-Lobatto-Legendre points (red).

Figure 3 . 6 :

 36 Figure 3.6: The vector field g( 0 , 1 , 2 )definedin(3.38)onthestandardtriangle. In this case ↵ =0.

Proposition 4 . 3 .

 43 Let ! 2 Λ k (T ) and consider a perturbation of !, e ! 2 Λ k (T ), such that ||! e !|| 0  ". One has

Figure 4 . 1

 41 Figure 4.1

Figure 4 . 3

 43 Figure 4.3Case k =2 : L e b e s g u ec o n s t a n t si nat r i a n g l eT ⇢ R 2 ,a s s o c i a t e dw i t hu n i f o r m (X 2 r,min (T )) and nonuniform (constructed using the simplicial isomorphism given in Equation(3.38) for ↵ =0and↵ = ↵ opt )distributionsofsmallfacesfordifferen t polynomial degrees r 1. Results are reported in Table4.4 on the left. A visual comparison in semi-logarithmic scale is offered in Figure4.3 on the right. Results associated with uniform simplices are depicted in red, whereas blue and green are associated with non uniform weights.

Figure 4 . 4 :

 44 Figure 4.4: Trends of the estimated generalised Lebesgue constants for the same set of nodes associated with k =0,k =1andk =2. Graphsareinsemi-logarithmic scale.

Figure 4 .

 4 5 on the right.

  k=2

Figure 4 . 6 1 .

 461 Figure 4.6 Case k =2: Lebesgueconstan tsinatetrahedronT ⇢ R 3 ,associatedwithuniform (X 2 r,min (T )) and nonuniform distributions of small faces for polynomial degrees r 1. Results are reported in Table 4.7 on the left. A visual comparison in semi-logarithmic scale is offered in Figure 4.6 on the right.

Figure 4 .

 4 7 on the right.

Figure 4 . 8 :

 48 Figure 4.8: Trends of the estimated generalised Lebesgue constants for the same set of nodes associated with k =0 ,k =1 ,k =2a n dk =3 . G r a p h sa r ei n semi-logarithmic scale.

4 p 6 .

 46 particular, one has | max | = 2 and | min | = W et h u sp l a c eo u r s e l v e si nt h e case of k =2 ,w h e n c e | max| 2 | min | 2 = 3 2 .T a b l e4 .

Algorithm 2 :

 2 The numerical estimation of stability zeronorm = zeros(deg,t) create a test mesh of k-simplices e C k of cardinality M for r = 1:deg consider the set of unisolvent and minimal small k-simplices S k r of cardinality N compute the Vandermonde matrix V = R si ! j by quadrature rule compute the matrix W = R ci ! j of size M ⇥ N by quadrature rule for `= 1:t put " = 10 create the random perturbation vector = " ⇤ (2 ⇤ rand(M, 1) 1) compute

uniform (X 1 r

 1 ,min (T )) non uniform (warp and blend) r Leb. c. " = 10 2 " = 10 5 " = 10 8 Leb. c. " = 10 2 " = 10 5 " = 10

( 4 . 1 " kΠ k r ! Π k r e !k 0 ◆Λ r 1 " kΠ k r ! Π k r e !k 0 ◆ = b 00 ,

 4101000 11) reproduces the behaviour of Λ r up to a multiplicative constant independent of r. Indeed, estimated Lebesgue constants and the left hand sides of (4.11) show a parallelism in semilogarithmic scale. Expanding, we get log Λ r = ar + b and log ✓ = ar + b 0 , for some a, b, b 0 2 R. Thus log ✓ for b 00 = b b 0 . Thus there exists k2 R such that Λ r ⇡ k 1 " kΠ k r ! Π k r e!k 0 , whence the claimed sharpness. To better appreciate this fact, we address toFigures 4.10

Figure 4 . 10 :

 410 Figure 4.10: Left, the case of k =0i nt h es t a n d a r dt r i a n g l eT ⇢ R 2 .R i g h t ,t h e case of k =0i nt h es a m et r i a n g l eT ⇢ R 2 .R e dl i n e sr e p r e s e n tr e s u l t sr e l a t e dt o uniform simplices, whereas blue lines represent results associated with warp and blend simplices. Dashed lines are the left hand side of (4.11), whereas straight lines the generalised Lebesgue constants computed in Section 4.2.

Figure 4 . 11 :

 411 Figure 4.11: The case of k =2i nt h es t a n d a r dt r i a n g l eT ⇢ R 2 .R e d l i n e s represent results related to uniform simplices, whereas blue lines represent results associated with warp and blend simplices. Dashed lines are the left hand side of (4.11), whereas straight lines the generalised Lebesgue constants computed in Table 4.4.

Figure 4 . 12 :

 412 Figure 4.12: Left, the case of k = 0 in the standard tetrahedron T ⇢ R 3 .R i g h t , the case of k =1i nt h es t a n d a r dt e t r a h e d r o nT ⇢ R 3 .R e d l i n e s r e p r e s e n t results related to uniform simplices, whereas blue lines represent results associated with warp and blend simplices. Dashed lines are the left hand side of (4.11), whereas straight lines the generalised Lebesgue constants computed in Table4.5 (left, k =0)andT able4.6(righ t,k =1).

Figure 4 .

 4 Figure 4.13: Left, the case of k = 2 in the standard tetrahedron T ⇢ R 3 .R i g h t , the case of k =3i nt h es t a n d a r dt e t r a h e d r o nT ⇢ R 3 .R e d l i n e s r e p r e s e n t results related to uniform simplices, whereas blue lines represent results associated with warp and blend simplices. Dashed lines are the left hand side of (4.11), whereas straight lines the generalised Lebesgue constants computed in Table4.7 (left, k =2)andT able4.8(righ t,k =3).

Figure 4 . 15 Left

 415 Figure 4.15 Left, in Table 4.16, behaviour of ||! Π 1 r !|| 0 for weights associated with different choices of small simplices in the standard tetrahedron T ⇢ R 3 .O n eh a s||!|| 0 ⇡ 0.2957. Right, in Figure 4.15, a visual representation of Table4.16 with respect to the polynomial degree r.I na c c o r d a n c ew i t ht h er e s u l to b t a i n e df o rT ⇢ R 2 , regression lines show a slow convergence for weights associated with non-uniform distribution of weights, in contrast with the uniform one.

  and

  see Remark 2.26. The result follows from Proposition 2.17. Theorem 2.28. The set X k r,min (T ) is unisolvent for P r Λ k (T ). Proof. For k = n, in view of Remark 2.26, this is just Lemma 2.27. Also, we have already observed that for k =0thisfollo wsfromthefactthatX 0 r,min (T )=X 0 r (T ) and for r =1t h i sf o l l o w sf r o mL e m m a2 . 4 ,s i n c eX k 1,min

  1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 2 9.6391 ⇥ 10 0 7.7369 ⇥ 10 0 9.6391 ⇥ 10 0 7.7369 ⇥ 10 0 3 3.3437 ⇥ 10 1 7.3193 ⇥ 10 1 2.6374 ⇥ 10 1 6.0605 ⇥ 10 1 4 1.6905 ⇥ 10 2 7.8838 ⇥ 10 2 7.1572 ⇥ 10 1 4.7589 ⇥ 10 2 5 9.9617 ⇥ 10 2 9.0257 ⇥ 10 3 2.0038 ⇥ 10 2 3.7485 ⇥ 10 3 6 5.3075 ⇥ 10 3 1.0711 ⇥ 10 5 5.6324 ⇥ 10 2 2.9610 ⇥ 10 4 7 3.3716 ⇥ 10 4 1.3019 ⇥ 10 6 1.5876 ⇥ 10 3 2.3442 ⇥ 10 5 8 2.1106 ⇥ 10 5 1.6091 ⇥ 10 7 4.4782 ⇥ 10 3 1.8590 ⇥ 10 6 9 1.3707 ⇥ 10 6 2.0135 ⇥ 10 8 1.2635 ⇥ 10 4 1.4763 ⇥ 10 7

Table 3 .

 3 10 0 2 1.1074 ⇥ 1.5202 ⇥ 10 1 9.5200 ⇥ 10 0 1.1074 ⇥ 10 1 9.2892 ⇥ 10 0 9.5199 ⇥ 10 0 3 7.1243 ⇥ 4.7937 ⇥ 10 1 1.1282 ⇥ 10 2 4.2973 ⇥ 10 1 5.2716 ⇥ 10 1 1.0513 ⇥ 10 2 4 4.3252 ⇥ 2.1377 ⇥ 10 2 1.5608 ⇥ 10 3 1.3742 ⇥ 10 2 3.1195 ⇥ 10 2 1.0381 ⇥ 10 3 5 3.4161 ⇥ 1.9776 ⇥ 10 3 2.3725 ⇥ 10 4 4.2761 ⇥ 10 2 2.4726 ⇥ 10 3 9.5848 ⇥ 10 3 6 2.3708 ⇥ 2.4115 ⇥ 10 4 3.8069 ⇥ 10 5 1.3035 ⇥ 10 3 1.7750 ⇥ 10 4 8.3650 ⇥ 10 4 7 1.9066 ⇥ 3.3117 ⇥ 10 5 6.3167 ⇥ 10 6 3.9360 ⇥ 10 3 1.2803 ⇥ 10 5 7.0108 ⇥ 10 5 8 1.5399 ⇥ 5.0095 ⇥ 10 6 1.0726 ⇥ 10 8 1.1751 ⇥ 10 4 1.0204 ⇥ 10 6 5.6925 ⇥ 10 6 9 1.3852 ⇥ 7.9969 ⇥ 10 7 1.8523 ⇥ 10 9 3.4779 ⇥ 10 4 9.4665 ⇥ 10 6 4.5067 ⇥ 10 7

	2: Condition numb er of the generalised Vandermonde matrix asso ciated
	with X k r,min

  and only if V is. By Proposition 3.2 we have that a set is unisolvent and minimal if and only if the associated generalised Vandermonde matrix is invertible, whence S k r (T )i su n i s o l v e n ta n dm i n i m a lf o rP r Λ k (T )i fa n d only if S k r (T ' )i su n i s o l v e n ta n dm i n i m a lf o rP r Λ k (T '

Table 4 .

 4 1: Case k =0: Lebesgueconstan tsinatriangleT ⇢ R 2 ,foruniformpoints (i.e. X 0 r (T )=L r (T )) and nonuniform ones obtained by computing warp and blend nodes for parameters ↵ =0and↵ = ↵ opt for different polynomial degrees r 1.

  .2. A visual comparison is provided in Figure 4.1.

		uniform	nonuniform
	r X 1 r,min (T ) Sym. Lob. W. & B.
	1	1.00	1.00	1.00
	2	4.94	4.94	4.94
	3	7.92	6.71	6.71
	4	12.17	8.16	8.16
	5	18.92	9.61	9.60
	6	29.95	11.80	11.62
	7	48.31	14.71	14.51
	8	79.45	18.13	17.65
	9	133.03	20.99	20.32	10 1
	10	226.20	28.74	24.44
	11	389.59	38.15	29.19	2	3	4	5	6 Polynomial degree 7 8	9	10	11	12
	12	678.10	52.97	35.85
			Table 4.2

  Figure 4.2: A non symmetric configuration of unisolvent and minimal 1-simplices in a triangle T ⇢ R 2 . Corresponding results are reported in Table 4.3.The associated Lebesgue constant are extremely discouraging, even compared with Lebesgue constants associated with uniform small 1-simplices. This is shown inTable 4.3.

		x 2
		0	x 1
	r Uniform Non sym. Lobatto
	1	1.00	1.00
	2	4.94	4.94
	3	7.92	6.67
	4	12.17	9.17
	5	18.92	14.51
	6	29.95	23.49
	7	48.31	41.55
	8	79.45	77.15
	9	133.03	154.18
	10	226.20	327.36
	11	389.59	827.80
	12	678.10	2142.45

Table 4 .

 4

	3: Case k = 1: Lebesgue constants in a triangle T ⇢ R 2 ,a s s o c i a t e dw i t h
	uniform (that is, X 1 r,min (T )) and non symmetrised edges based on Gauss-Lobatto-
	Legendre points, as in Figure 4.2.

  .1.

		uniform	nonuniform
	r X 2 r,min (T ) Sym. Lob. W. & B.
	1	1.00	1.00	1.00
	2	2.53	2.53	2.53
	3	4.93	3.82	3.82
	4	8.72	4.86	4.86
	5	14.77	5.63	5.63
	6	24.42	6.91	6.57
	7	39.79	7.88	6.64
	8	64.22	8.99	7.33
	9	102.86	9.57	8.39
	10 11	163.59 258.30	11.40 14.21	10.41 12.11	10 0	2	3	4	5	Polynomial degree 6 7 8	9	10	11	12
	12	404.59	26.54	25.87
			Table 4.4

Table 4 .

 4 5: Case k =0 : L e be s g u ec o n s t a n t si nat e t r a h e d r o nT ⇢ R 3 ,f o ru n i f o r m points (i.e. L r (T )) and nonuniform ones obtained by computing warp and blend nodes for parameters ↵ =0and↵ = ↵ opt for different polynomial degrees r 1.

		uniform	nonuniform
	r X 0 r,min (T ) Sym. Lob. W. & B.
	1	1.00	1.00	1.00
	2	2.00	2.00	2.00
	3	3.01	2.93	2.93
	4	4.84	4.03	4.03
	5	7.88	5.25	5.24
	6	13.66	7.34	6.98
	7	22.61	9.47	8.87
	8	37.62	12.83	12.30
	9	69.30	18.38	16.73

Table 4 . 6

 46

	Lebesgue constant: k = 1
	Lebesgue c. Unif
	Lebesgue c. SL
	Lebesgue c. W & B
	10 2
	log (value)
	10 1
	23456789
	Polynomial degree
	Figure 4.5

  .6. A visual representation of Table4.7 is provided in Figure4.6.

		uniform	nonuniform
	r X 2 r,min (T ) Sym. Lob. W. & B.
	1	1.00	1.00	1.00
	2	7.51	7.51	7.51
	3	13.04	12.28	12.28
	4	22.24	19.78	19.78
	5	35.79	27.01	27.01
	6	55.07	33.66	32.87
	7	83.40	41.86	37.70
	8	120.84	48.00	42.00
	9	190.02	58.90	51.28

Table 4 . 7

 47

		Lebesgue constant: k = 2
		Lebesgue c. Unif
		Lebesgue c. SL
	10 2	Lebesgue c. W & B
	log (value)
	10 1
		23456789
		Polynomial degree

  .7, left, for what concerns uniform small simplices X 3 r,min and right for what concerns warp and blend simplices.

		uniform	nonuniform		10 2	Lebesgue constant: k = 3
	r X 3 r,min (T ) Sym. Lob. W. & B.			Lebesgue c. Unif Lebesgue c. SL Lebesgue c. W & B
	1	1.00	1.00	1.00
	2	2.37	2.37	2.37
	3 4	4.39 7.43	3.45 4.44	3.45 4.44	log (value)	10 1
	5	12.11	5.58	5.40
	6	19.34	6.95	6.07
	7	30.59	8.04	7.54
	8 9	48.05 76.58	9.87 14.90	9.46 13.95		10 0	Polynomial degree 23456789
			Table 4.8				Figure 4.7

Case k =3: Lebesgueconstan tsinatetrahedronT ⇢ R 3 ,associatedwithuniform (X 3 r,min (T )) and nonuniform distributions of small faces for polynomial degrees r 1. Results are reported in Table

4

.8 on the left. A visual comparison in semi-logarithmic scale is offered in

  One has 2 (M )=2 . W en o wc o n s i d e ru n i f o r m ,s y m m e t r i s e dL o b a t t oa n dw a r p and blend simplices on T ' . As test mesh, we consider the image of the test mesh for T under the action of '. For each choice of set of simplices, we show quantities

							1
							A .
							1 p 6
	Λr(Tϕ) Λr(T ) and compare them with the theoretical bound k 2 (M ). Results for k =1 and k =2aregiv eninT able4.9.
			k =1				k =2
		uniform	nonuniform	2 (M ) uniform	nonuniform	2 2 (M )
	r X 1 r,min	Sym. Lob W. & B.		X 2 r,min	Sym. Lob. W. & B.
	3	1.0098	1.0039	1.0039	2.0000	1.1139	1.1139	1.1139	4.0000
	4	1.0654	1.1317	1.1317	2.0000	1.0730	1.0878	1.0878	4.0000
	5	1.0850	1.2264	1.1699	2.0000	1.1217	1.1266	1.1265	4.0000
	6	1.0924	1.2858	1.1387	2.0000	1.2020	1.1796	1.1775	4.0000
	7	1.1420	1.2644	1.1723	2.0000	1.2790	1.2753	1.2815	4.0000
	8	1.1146	1.2791	1.2028	2.0000	1.4279	1.4476	1.4844	4.0000

  1 0r e p o r t s Λr(Tϕ) Λr(T ) for warp and blend simplices up to degree r =10sho wingthattheyexceed | max| 2 | min | 2 .

	r W. & B. | max | 2 /| min | 2
	7	1.2815	1.5000
	8	1.4844	1.5000
	9	1.4961	1.5000
	10	1.5234	1.5000

Table 4 .

 4 10: For k =2,r = 10, the ratio of the Lebesgue constant associated with warp and blend simplices computed on T ' and that on T exceeds | max | 2 /| min | 2 .

Table 4 .

 4 1 3 ,s h r i n k i n g the base of T to a point. Results are reported in Table4.11 and confirm the theoretical estimate.

		uniform	nonuniform
		X 1 7 (T ) Sym. Lob. W. & B.	2 (M )
	1/6	1.1420	1.2644	1.1723	2
	3/12	1.9612	2.0309	1.8931	4
	7/24	3.7344	3.7769	3.5219	8
	15/48	7.3611	7.3468	6.8460	16
	31/96 14.6565	14.5256	13.5295	32
	11: The quantity 2 and the ratio Λ 1 7 (T )/Λ 1 7 (T )whenT is the tetrahedron
	of

  Figure 4.9: Supports obtained by linearity for Σ 2 3 (T )b e i n gT =[ x 0 , x 1 , x 2 ]a2simplex: three triangles and three diamonds. Higher dimensional examples bear more complex shapes.

			x 0
	x 1			x 2
		simplicial	non simplicial
	r X 2 r,min (T )Σ 2 r,min (T )	Tiles
	2	7.51	28.17	12.67
	3	13.04	114.11	40.56
	4	22.24	329.78	100.62
	5	35.79	781.88	214.92
	6	55.07	1635.72	433.61
	Table 4.13: Leb esgue constants for k =2inatetrahedron T ,a s s oc i a t e dw i t ht i l e s
	built upon uniform distributions and compared with the corresponding simplices
	in X 2 r,min (T )a n dΣ 2 r,min (T ).
	4.3 Stability results: verification of the Lebesgue
	constants
	Proposition 4.3 offers a result about stability.I n d e e d , i t a l l o w s t o b o u n d t h e
	propagation of perturbation e ! that one may have on ! using the Lebesgue constant
	as	1 "	kΠ
				2 ),
	for different values of 2 [0, 1/3) and different distributions of nodes.

Table 4 .

 4 14: A comparison of the left (columns 3 5and7 9) and the right hand side (columns 2 and 6) of Equation (4.11) for uniform (columns 2 5) and warp and blend (columns 6 9) simplices. Orders of perturbation are " =10 2 , 10 5 , 10 8 . This table refers to the case k =1inatriangleT ⇢ R 2 .

	8

Remark 4.5. These numerical experiments show that the estimate provided by Proposition 4.3 is sharp, in the sense that the left hand side of

sometimes called simplexes, especially in older references.

not to be confused with the map ⌧ ξ of[START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF], which is here denoted by ξ .

Acknowledgements

the cardinal basis. We have ω s = V T ω = V T P 1 η =( PV T ) 1 η. Thus ! s i = P N k=1 (PV T ) 1 i,k ⌘ k . Following (3.6), we have that V 0 = VP T , whence

Plugging this into (3.19) we get the claim.

Properties (3.16) and (3.17) can be extended to the generalised Lebesgue constant. Higher dimensional counterparts will be given in Proposition 4.3 and (4.5).

Dependence on the simplex

We have shown that for k =0t h eg e n e r a l i s e dL e b e s g u ec o n s t a n tb o i l sd o w nt o the usual nodal Lebesgue constant. Such a quantity reads the positioning of nodes inside the domain [START_REF] Hesthaven | From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex[END_REF]. Under the theoretical point of view, this is not affected by the domain itself, as we shall make clear later. Interestingly, this does not extend to k>0, except when k = n.T os e et h i s ,w ew o r ka sf o l l o w s .

Let T be an n-simplex generated by {x 0 ,...,x n } and ' be a non degenerate affinity, i.e. a map

with b 2 R n and det M 6 =0. ThenT ' . = '(T )isanothern-simplex of R n generated by {'(x 0 ),...,'(x n )}.T h er e l a t i v ep o s i t i o n( w i t hr e s p e c tt ov e r t i c e so fT )o fa point x 2 T coincides with that of '(x) 2 T ' (with respect to vertices of T ' ), i.e. if x = P i x i ,o n eh a s'(x)= P i '(x i ). We show that a non degenerate affinity maps unisolvent and minimal sets to unisolvent and minimal sets. Lemma 3.10. S k r (T )={s 1 ,...,s N } is a unisolvent and minimal set for P r Λ k (T ) if and only if S k r (T ' )={'(s 1 ),...,'(s N )} is a unisolvent and minimal set for

Since ' is a non degenerate affinity, its pullback maps such a basis into a basis {' ⇤ ! ' 1 ,...,' ⇤ ! ' N } for P r Λ k (T ), see Equation (1.18) in Proposition 1.11. Let V ' and V denote the generalised Vandermonde matrices with resp ect to

We proved that the generalised Leb esgue constant for n-forms is independent on the choice of T .

Note that a central role in the proof of Proposition 3.14 is played by the fact that n-volumes in R n are transformed accordingly to the determinant of M .T h i s becomes false when k 6 = n.T oh a v ea ne x a m p l e ,c o n s i d e ri nR 3 the oblique face F 0 of the standard 3-simplex T ,a n da na ffi n i t yr e p r e s e n t e db yt h em a t r i x

2 and det M =1,whereas|'(F 0 )| 0 = 1 2 .I ng r e a tg e n e r a l i t y ,t h i s is due to the fact that the k-volume of a k-simplex T =[ v 1 ,...,v k+1 ]i nR n is computed as

the so-called Cayley-Menger determinant [START_REF] Berger | Geometry I[END_REF]. D is defined as the matrix

being e D the (k +1 )⇥ (k +1 ) m a t r i x w h o s e ( i, j)-th entry is kv i v j k 2 2 ,t h e Euclidean distance between the i-th vertex v i of T and the j-th v j (see [START_REF] Christiansen | On high order finite element spaces of differential forms[END_REF], where D is given the name of augmented matrix ). The complexity of handling this quantity when k 3h a sb e e nr e c o g n i z e di n[ 2 6 ] . T h i sf o r c e su st oc h a n g e the approach when 0 <k<n.I nf a c t ,a s s u m i n gt h a tD ' is the matrix associated with {'(v 1 ),...,'(v k+1 )},i ti sn o tk n o w n ,i ng e n e r a l ,aw a yt oe x p r e s sn e a t l yi t s determinant in terms of the product of det D and some quantity related to M .

We are thus forced to adopt another reasoning and try to b ound the deformation of the k-volume in terms of some quantities of M .T h e s ee s t i m a t e s ,a l o n g with the result of Corollary 3.12, yield bounds for the generalised Lebesgue constant. To fix ideas and produce a convincing guess for the most general case, we first assume that M is diagonal. This makes it easier and more intuitive to bound transformations of volume. This restrictive assumption gives in fact in turn a clear estimate, which is presented in Equation (3.28), that suggests a meaningful generalisation. Reproducing techniques inherited from the diagonal case we shall then prove it.

Let | max | and | min | denote, respectively, the largest and smallest (in magnitude) eigenvalues of M .S u c hq u a n t i t i e sr e p r e s e n th o w ,a tm o s ta n da tl e a s t ,a k-simplex is stretched under the action of ',a sn e x tr e s u l ts h o w s . Lemma 3.15. Let '(x)=M x + b be a non degenerate affinity and assume M is diagonal. Let e =[x, y] and '(e) be its transformed under '. One has

Proof. We first show the rightmost inequality in (3.25). We have

The other inequality of (3.25) is obtained in the same way. We now prove the rightmost inequality of (3.26). Consider a k-cube C.S i n c e its volume is the product of the length of its sides, we may apply (3.25) to each of its edges to deduce that |C| 0 may be bounded as

We thus compute the volume of the k-simplex T as a countable sum of k-cubes

The other inequality follows from an analogous reasoning.

We may thus prove the following.

Chapter 3

Figure 3.7: The action of the simplicial isomorphism ' induced by g on X 3 (T ).

Left, the complex X 3 (T )a n dr i g h t ,t h ec o m p l e xY 3 (T ) . = ' (X 3 (T )) for ↵ = ↵ opt .

↵ = ↵ opt , that was studied in [START_REF] Hesthaven | Nodal discontinuous Galerkin methods[END_REF] and which we refer to as warp and blend simplices.

Their performances have been outlined in [3] for k =1a n di n [ 2 ]f o rk =2 . T h e following tables report the condition numbers of the generalised Vandermonde matrices related to small simplices associated with both ↵ =0andtheoptimised ↵ = ↵ opt with respect to the Bernstein basis B α ! F defined in Equation (3.14).

In particular, Table 3.3 contains condition numbers for symmetrised Lobatto and warp and blend simplices for n =2 ,w h e r e a sT a b l e3 . 4t h o s ef o rs y m m e t r i s e d Lobatto and warp and blend simplices for n =3. Optimisedv alues↵ opt are taken from [START_REF] Warburton | An explicit construction of interpolation nodes on the simplex[END_REF] and [START_REF] Hesthaven | Nodal discontinuous Galerkin methods[END_REF].

We have thus shown that the generalised Vandermonde matrices asso ciated with these sets of simplices are non degenerate, hence the corresponding collections of weights are minimal and unisolvent for P r Λ k (T ). It is worth noting that there is not a very significant improvement in conditioning with respect to values obtained for uniform simplices (see Table 3.1 for two dimensional and Table 3.2 for three dimensional results). As we shall see in next chapter, this is in stark contrast with performances of the associated sets of weights for what concerns the relative generalised Lebesgue constants.

Symmetrised Lobatto: ↵ =0 Warp and blend:

1 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 10 0 1.0000 ⇥ 1 8 ] . I ne l e c t r o m a gnetism, in fact, smooth 1-forms and 2-forms in R 3 represent, respectively, the electric and the magnetic field. These quantities can be observed and read as circulation and fluxes. Weights offer suitable degrees of freedom for these entities, since they can be measured. Hence the left hand side of (4.3) is given and thus its interpolation onto P r Λ k (T ), being V computable as in Chapter 3. However, any physical quantity that is obtained by a measurement contains an error and is subjected to perturbation. In view of (4.6), reducing the Lebesgue constant, as in the nodal case, allows to bound the propagation of such uncertainties.

Ap h y s i c a lm o t i v a t i o nt oP r o p o s i t i o n4 . 3i sp r o v i d e db y[

Computation of Lebesgue constants

The importance of computing the generalised Lebesgue constant has been addressed in previous chapters and made precise in (4.5) and (4.6). Recall that it is defined as

Of course, we are not able to compute it exactly since the supremum appearing in the definition ranges over all the possible k-chains supported in T .F o rt h i sr e a s o n we estimate it by considering a sufficiently large collection of test k-simplices; of course, this is a lower bound for Λ r ,b e i n gt h i sc o l l e c t i o nc o n t a i n e di nC k (T ). Notice that the finiteness of the collection allows to compute such a supremum on simplices and not chains. However, exploiting (4.6), we will show in the next section that this estimate is trustful. Various computations for k =1h a v eb e e n done in [3] and for k = 2 in [2]. An algorithmic scheme is the following. for C(S 1 r )=max j M y j x j ky j x j k 2 2

.W et h u sc o n s i d e ra sS 1 r the set of uniform small simplices X 1 7 (T ), so that either y j x j ky j x j k 2 =(1, 0) T , y j x j ky j x j k 2 =(0, 1) T or

⌘ T .W er e p o r tt h ec o r r e s p o n d i n gr e s u l t si nT a b l e4 . 1 2 . Table 4.12: A comparison b etween Λ 7 (T δ ) Λ 7 (T ) and C(X 1 7 (T ))kM 1 k 2 . T is the tetrahedron of vertices (0, 0, 0), (1 2 , , ), ( , 1 2 , ), ( , , 1 2 )consideringdifferent values of 2 [0, 1/3).

Non simplicial supports

Exploiting the linearity of integral with respect to the domain, one may convert the set Σ k r,min (T )intoacollectionofunisolventandminimal(triviallybyconstruction) supports that do not overlap. In this case, when k =1o n er e t r i e v e st h es t r a t e g y of chopping presented in Theorem 2.29. On the contrary, for k>1o n ed o e s not obtain only simplices, but other geometrical shapes, polyhedra in general, as Figure 4.9 shows.

We refer to these ob jects as tiles,s i n c et h e yp a v et h en-simplex T without overlapping, as depicted in Figure 4.9. Numerical results here reported discredit this choice, in view of badly performing Lebesgue constants, as reported in Table 4. [START_REF] Blyth | A Lobatto interpolation grid over the triangle[END_REF]. In that same table we analyse the role of overlappings, testing performances of simplices Σ k r,min (T )i n t r o d u c e di nS e c t i o n2 . 4 . T h e yo ff e rt h el a r g e s tL e b e s g u e constants observed in this work; for this reason we shall only consider them as a useful theoretical tool.

In particular, we see that tiles improve the results obtained with overlapping as in Σ ci ! sj by quadrature rule interpolate on e C k as q = W \ w measure norm(r, 1)=norm(q v, 0 inf 0 ) end uniform nonuniform r X 4.15 with respect to the polynomial degree r. Regression lines show a slow convergence for non-uniform weights, in contrast with uniform ones.

List of Figures

List of Tables