Le thèse est un projet qui exige de l'endurance, de l'engagement et de la patience.

A travers les quatre dernières années que je suis un doctorant à UPMC, je rencontrais des gens qui m'aidé avec la charge émotionnelle et scientifique qui vient avec un travail de ce genre. À la fin de ce travail, je tiens à exprimer ma gratitude et à adresser quelques mots à tous ceux qui, de près ou de loin, ont été présents et ont contribué à sa réalisation. Je tiens tout d'abord à exprimer ma profonde reconnaissance envers Christine BERNARDI. Ces années de thèse sous sa direction furent pour moi un vrai enrichissement. Je le remercie d'avoir accepté d'encadrer cette thèse. J'ai pu bénéficier de son intuition mathématique. Je voudrais le remercier pour tous ses encouragements constants, sa disponibilité et sa profonde gentillesse qui m'ont permis d'avancer sereinement dans mon travail. Je n'oublierai jamais son soutien et sa disponibilité dans les moments de doute. Juger un travail de thèse n'est pas une chose aisée. Pour cela, je suis extrêmement honorée que Madame Florence Hubert et Monsieur Mejdi Azaiez aient accepté de consacrer du temps à lire ce mémoire et à rédiger un rapport. Ils ont également contribué par leurs remarques et suggestions à améliorer la qualité de ce mémoire, et je les en remercie. Je leur suis également très reconnaissante d'avoir pris la peine de se déplacer pour assister à ma soutenance. Dedicace Je tiens à remercier très vivement M. Yvon Maday qui me fait l?honneur de présider mon jury de thèse. Je souhaiterais remercier le Professeur Faker Ben Belgacem qui a eu la gentillesse de me conseiller, pour le temps qu'il m'a accordé à plusieurs reprises au cours de cette thèse ainsi que pour l'intérêt qu'il a porté à mes travaux. J'adresse une pensé particulière à Khashayar Dadras et Christian David pour son sourire, sa disponibilité, sa gentillesse et son amitié. Je le remercie également pour l'impression de ce manuscript malgré son état de santé. Je remercie Antoine Le Hyaric pour les nombreuses fois où il m'a aidée avec toutes sortes de problèmes (logiciels scientifiques ou autres) ainsi que pour son amitié et sa gentillesse à mon égard. Je voudrais remercier mes collègues dans bureau 1516-316 qui m'ont accompagné pendant ces années de ma thèse à Jussieu, j'ai connu des nouvaux collègues dans bureau 1516-303 qui je remercie pour les nombreuses fois où il m'a aidé. Je voudrais remercier pour ces années passées au laboratoire Jacques-Louis Lions tous ses membres pour la bonne humeur, l'atmosphère à la fois studieuse. Je souhaiterais remercier mes rapporteurs pour le temps qu'ils ont accordé à la lecture de cette thèse et à l'élaboration de leur rapport. Je remercie mes amis chinois de Paris pour l'ambiance amicale qu'ils ont créée, je pense à Ma Yue, Zhu Jiamin, Guo Hao et bien d'autres. Je remercie particulièrement Yang Xingdi de m'avoir beaucoup aidé et accompagné. Merci à lui pour les moments agréables que nous avons passés ensemble à Paris. Merci pour son soutien et pour tout ce qu'elle a fait pour moi. Mes derniers remerciements iront évidemment à mon famile.

Chapter 1 Introduction

Motivation

In this thesis, we mainly talk about the problem of the flow of a viscous incompressible fluid moving in a rigid porous inhomogeneous medium. A good example of this problem is that water moves in the subsurface.

In the subsurface, rock is deposited in layers. Fluid flow within and between the rock layers is governed by the permeability of the rocks. However, to account for permeability, it must be measured in both the vertical and horizontal directions. For example, shale typically has permeabilities that are much lower vertically than horizontally (assuming flat lying shale beds). This means that it is difficult for fluid to flow up and down through a shale bed but much easier for it to flow from side to side.

Ultimately, if the pressure difference between a hydraulically fractured zone and a fresh water aquifer is not great, the distance between the zones is relatively large, and there are rocks with low vertical permeabilities in between the deeper and the shallower zones, flow between the zones is unlikely to occur. The exception to this is where there is a separate flow pathway such as an open bore hole or a series of faults or joints that intersect both the fractured zone and the fresh water aquifer.

Under either of these circumstances, the pressure difference and distance will be the determining factors as to whether fluid can migrate from the lower to the upper zone.

The principle that governs how fluid moves in the subsurface is called Darcy's law.

Darcy's law is an equation that defines the ability of a fluid to flow through a porous media such as rock. The law was formulated by Henry Darcy based on the results of experiments [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF] on the flow of water through beds of sand. It also forms the scientific basis of fluid permeability used in the earth sciences, particularly in hydrogeology. Although Darcy's law (an expression of conservation of momentum) was determined experimentally by Darcy, it has since been derived from the Navier-Stokes equations via homogenization. It is analogous to Fourier's law in the field of heat conduction, Ohm's law in the field of electrical networks, or Fick's law in diffusion theory.

One application of Darcy's law is to water flow through an aquifer; Darcy's law along with the equation of conservation of mass are equivalent to the groundwater flow equation, one of the basic relationships of hydrogeology. Darcy's law is also used to describe oil, water, and gas flows through petroleum reservoirs.

Darcy's law at constant elevation is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance. Darcy found that his data could be described by

U = kA µ P b P a L , (1.0.1)
where, U (units of volume per time, e.g., m 3 /s) is equal to the product of the intrinsic permeability of the medium, k(m 2 ), the cross-sectional area to flow, A (units of area, e.g., m 2 ), (P b P a ) (Pascals) is the total pressure drop, and L, the length over which the pressure drop.

The negative sign is needed because fluid flows from high pressure to low pressure.

Note: the elevation head must be taken into account if the inlet and outlet are at different elevations. If the change in pressure is negative (where P a > P b ), then the flow will be in the positive 0 x 0 direction. Dividing both sides of the equation by the area and using more general notation leads

u = k µ rP, (1.0.2)
where, u is the flux (discharge per unit area, with units of length per time, m/s) and rP is the pressure gradient vector (P a/m).

For very short time scales, a time derivative of flux may be added to Darcy's law, which results in valid solutions at very small times (in heat transfer, this is called the modified form of Fourier's law),

⌧ @u @t + u = k µ rh, (1.0.3)
where ⌧ is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times.

At last, in fact the water is the incompressible fluid, so the equation will also satisfy the incompressible condition:

r • (u) = 0.
A long this thesis we will focus on the following model : find a velocity u and pressure p solutions of :

8 > > > > > > < > > > > > > : ↵u + grad p = ↵f in ⌦, div u = 0 in ⌦, u • n = k o n 1 , p = p 0 on 2 (1.0.4)

Spectral, spectral element and mortar element methods

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, often involving the use of the Fast Fourier Transform. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

Spectral methods have been used extensively during the last decades for the numerical solution of partial differential equations (PDE) due to their bigger accuracy when compared to Finite Differences (FD) and Finite Elements (FE) methods. The rate of convergence of spectral approximations depends only on the smoothness of the solution, yielding the ability to achieve high precision with a small number of data.

This fact is known in literature as "spectral accuracy".

Spectral element techniques are high order methods which allow for either obtaining very accurate results or reducing the number of degrees of freedom for a fixed standard accuracy. As firstly explained by [START_REF] Steven | Comparison of pseudospectral and spectral approximation[END_REF][START_REF] Gottlieb | Numerical analysis of spectral methods[END_REF], they rely on high degree piecewise polynomial approximation, and on the use of tensorized bases of polynomials.

For these reasons, the basic geometries for these methods are tensorized, i.e. they are square or cubes. We refer to [START_REF] Funaro | Polynomial approximation of differential equations[END_REF][START_REF] Hussaini | Spectral methods in fluid dynamics[END_REF][START_REF] Bernardi | Spectral methods[END_REF] for a general presentation and analysis of the spectral methods in these geometries, and also to [START_REF] Dubiner | Spectral methods on triangles and other domains[END_REF][START_REF] Cohen | Higher order triangular finite elements with mass lumping for the wave equation[END_REF][START_REF] Sherwin | Tetrahedral hp finite elements: Algorithms and flow simulations[END_REF] for extensions to triangles or tetrahedra. The discrete spaces for spectral methods are simply polynomial spaces, and the discretization parameter is the maximal degree of these polynomials.

The mortar element method, introduced in [START_REF] Bernardi | A new nonconforming approach to domain decomposition: the mortar element method[END_REF], is a domain decomposition technique which allows for working on general partitions of the domain without conformity restrictions. It is particularly important when combined with spectral-type discretizations, since handling complex geometries from very simple subdomains can be performed with this method in a very efficient way. It can also be used to couple different kinds of variational discretizations on the subdomains, such as finite elements or spectral methods. So it leads to discrete problems which are most often non-conforming in the Hodge sense, which means that the discrete space is not con-tained in the variational one. It was first analysed in the case of the 2D Laplace equation [START_REF] Bernardi | A new nonconforming approach to domain decomposition: the mortar element method[END_REF] which admits a natural variational formulation in the usual Sobolev space H 1 (⌦) of functions with square-integrable first-order derivatives. We also refer to [START_REF] Faker | Discrétisations 3D non conformes par la méthode de décomposition de domaine des éléments avec joints: Analyse mathématique et mise en oeuvre pour le probleme de Poisson[END_REF] for the first 3D results. It was extended [START_REF] Belhachmi | Méthodes d'éléments spectraux avec joints pour la résolution de problèmes d'ordre quatre[END_REF] to the bilaplacian equation where the variational space is the standard space H 2 (⌦) of functions with square-integrable first-order and second-order derivatives and also to the Stokes problem which is of saddle-point type; however, it still involves the usual Sobolev spaces. We also quote [START_REF] Belhachmi | Spectral element discretization of the circular driven cavity. part III: The Stokes equations in primitive variables[END_REF] for an application of the mortar technique to weighted Sobolev spaces, in order to handle discontinuous boundary conditions for the Navier-Stokes equations.

Objectives of the thesis

In this thesis, we mainly talk about the analysis of the mortar spectral element discretization of the problem (5.1.1 where ⌦ be a bounded, connected, open set in R d , d = 2 or 3, with a Lipschitz continuous boundary @⌦, and let n denote the unit outward normal vector to ⌦ on @⌦, 1 and 2 be a partition of @⌦ without overlop. ↵ is constant or piecewise continuous coefficients. The data are now the function f and the boundary conditions k and p 0 . The unknowns are the velocity u and the pressure (or hydraulic head, according to the model) p. we are interested in the case where this function is not globally continuous but only piecewise smooth and also such that the ratio of its maximal value to its minimal value is large. This models, for instance, the flow of a viscous incompressible fluid in a rigid porous inhomogeneous medium.

A very interesting and unusual feature of Darcy's equations is that they admit two equivalent variational formulations, whether the space of velocities is more regular than the space of pressures or not, see [START_REF] Azaïez | The mortar spectral element method in domains of operators. part I: The divergence operator and Darcy's equations[END_REF][START_REF] Achdou | A priori and a posteriori analysis of finite volume discretizations of Darcy's equations[END_REF][START_REF] Bernardi | Mortar finite element discretization for the flow in a nonhomogeneous porous medium[END_REF]. The drag coefficient ↵ will be considered as piecewise constant or piecewise smooth functions. The variational spaces can be choosen as (u; p) in H(div; ⌦) ⇥ L 2 (⌦) or in L 2 (⌦) ⇥ H 1 (⌦). So two mortar discrete problems based on Galerkin numerical integration (GNI) will be constructed.

Then we obtain the well-posedness and regularity properities of the problems. Next, it is necessary to prove a priori error estimates of spectral type: the order of convergence only depends on the regularity of the solution, more precisely on its local regularity in each subdomain. For the implementation of the mortar technique, it mainly relies on an appropriate treatment of the matching conditions on the interfaces.

In this thesis, we will also develop a spectral method for fourth-order differential equations in one dimension. A Legendre Petrov-Galerkin method for linear fourth-order differential equations and a Legendre Petrov-Galerkin and Chebyshev Collocation method for the nonlinear Kuramoto-Sivashinsky equation will be presented. Also we will prove the optimal rate of convergence in L 2 norm of the method, and numerical experiments will be given which demonstrate the efficient of proposed schemes.

Thesis outline

The thesis is composed by 2 parts. The first part is dedicated to the main subject of my work and concerns the numerical analysis of the martar method applied to the Darcy problem. During the first part of my thesis I worked on the analysis of a Legendre Petrov-Galekin method for linear fourth-order differential equations in one dimension and a Legendre Petrov-Galerkin and Chebyshev collocation method for the nonlinear Kuramoto-Sivashinsky equation have been developed. The second part contains the description of this contribution This invesgation will be given in the end of then mauscrit (see Appendix).

After a short introduction given in chapter one, we will consider in chapter 2 the Darcy's equations with general boundary conditions and piecewise continuous coefficients in a bounded domain. This problem can be formulated in two different spaces.

Then we prove the well-posedness of the equivalent variational problem of the Darcy's equation. We also give the regularity properties.

In chapter 3 we present the discretization of the steady Darcy problem. We propose spectral discretizations of this problem based on the Galerkin with Numerical Integration (G-NI) variants. The Numerical analysis of the discrete problem is performed.

We also present the two-dimensional numerical experiments.

In chapter 4 we consider Darcy's equations with general boundary conditions and piecewise continuous coefficients in a bounded domain. This problem can be formulated in spaces of square-integrable functions with square-integrable divergence. We propose a spectral element discretization of this problem which relies on the mortar domain decomposition technique. The Numerical analysis of the discrete problem is performed. We also present the two-dimensional numerical experiments. They turn out to be in good coherency with the theoretical results.

In chapter 5 we also consider Darcy's equations with general boundary conditions and piecewise continuous coefficients in a bounded domain. A spectral element discretization of this problem relies on the mortar domain decomposition technique is proposed.

We present the numerical analysis and two-dimensional numerical experiments, which turn out to be in good coherency. This chapter ends with the comparison of the two spectral element methods will be presented in Chapter 6.

Chapter 2

Darcy's Equations

Introduction

We first talk about the analysis of the mortar spectral element discretization of the problem introduced by Darcy [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF] as follows

8 > > > > > > < > > > > > > : ↵u + grad p = ↵f in ⌦, div u = 0 in ⌦, u • n = k o n 1 , p = p 0 on 2 (2.1.1)
where ⌦ is a bounded, connected, open set in R d , d equals to 2 or 3, with a Lipschitz continuous boundary @⌦, and n denotes the unit outward normal vector to ⌦ on @⌦, 1 and 2 , with a positive measure, being a partition of @⌦ without overlop. The data are now the equation f and the boundary conditions k and p 0 . The unknowns are the velocity u and the pressure (or hydraulic head, according to the model) p.

The coefficients ↵ are constant or piecewise continuous. We are interested in the case where this function is not globally continuous but only piecewise smooth and also such that the ratio of its maximal value to its minimal value is large. This models, for instance, the flow of a viscous incompressible fluid in a rigid porous inhomogeneous medium.

Sobolev spaces

In this section, we recall the main notions and results, concerning the classical Sobolev spaces, which will be used in later sections, also some properties. Although they are stated without proof, these results are complete, rigorous and fairly general.

We define D(⌦) to be the linear space of infinitely differentiable functions, with compact support on ⌦. Then we set

D( ⌦) = { | ⌦ ; 2 D(R d )} or equivalently, if O denotes any open subset of R N such that ⌦ ⇢ O,
Now, let D 0 (⌦) denote the dual space of D(⌦), often called the space of distributions on ⌦. We denote by < •, • > the duality pairing between D 0 (⌦) and D(⌦) and we remark that when f is locally integrable function, then f can be identified with a distribution by

< f, >= Z ⌦ f (x) (x)dx, 8 2 D(⌦)
In other words, < •, • > is an extension of the scalar product of L 2 (⌦).

For v = (v 1 , v 2 , ..., v N ), we define the divergence operator by:

div v = N X i=1 (@v i /@x i ).
Note the identity:

div (gradv) = 4v.
So far, we have been mainly interested in subspaces of H 1 (⌦) N ; but subsequently, it will be worthwhile to use functions with less regularity. Bearing this in mind, we introduce the following spaces:

H(div ; ⌦) = {v 2 L 2 (⌦) N ; div v 2 L 2 (⌦)},
which is clearly a Hilbert space for the norm: 1. There exists a unique linear continuous map 0 :

||v|| H(div ;⌦) = (||v|| 2 0,⌦ + ||div v|| 2 0,⌦ ) 1/2 . ( 2
H s (⌦) ! H s 1/2 ( ) such that 0 v = v| for each v 2 H s (⌦). 2.
There exists a linear continuous map R 0 :

H s 1/2 ( ) ! H s (⌦) such that 0 R 0 ' = ' for each ' 2 H s 1/2 ( ).

The next theorem concerns the normal component of boundary values of functions of

H(div ; ⌦).

Theorem 2.2

The mapping n : v ! v • n| defined on D( ⌦) N can be extended by continuity to a linear and continuous mapping, still denoted by n , from H(div ; ⌦)

to H 1/2 ( ).
By extension, n v is called the normal component of v on and is denoted simply by v • n.

From the above two theorem, we derive the following Green's formula:

(v, grad ) + (div v, ) =< v • n, > , 8v 2 H(div ; ⌦), 8 2 H 1 (⌦). (2.2.2)
As a consequence, we can now extend Green's formula for the Laplace operator to a wider range of functions.

Corollary 2.1 Let u 2 H 1 (⌦) and 4u 2 L 2 (⌦). Then @u/@n 2 H 1/2 ( ) and

(grad u, grad v) = (4u, v)+ < @u/@n, v > , 8v 2 H 1 (⌦). (2.2.3)
Two useful applications of Green's formula will also be given here.

Lemma 2.1 Let ⌦ be a bounded open subset of R N with a Lipschitz-contunuous boundary .

1. For u and v in H 1 (⌦) and for 1  i  N , we have

Z ⌦ u(@v/@x i )dx = Z ⌦ (@u/@x i )vdx + Z 0 u 0 vn i ds (2.2.4) 2. If in addition u 2 H 2 (⌦) we have N X i=1 Z ⌦ @u @x i @v @x i dx = N X i=1 Z ⌦ @ 2 u @x 2 i + N X i=1 Z ( @u @x i v)n i ds (2.2.5)
Adopting the usual notations

4u = N X i=1 @ 2 u @x 2 i
, gradu = (@u/@x 1 , ...@u/@x N )

(2.2.5) becomes:

(gradu, gradv) = (4u, v) + Z (@u/@n) 0 vds. (2.2.6)
When ⌃ is an open (non-empty) Lipschitz continuous subset of the boundary , we introduce the space

H 1 ⌃ (⌦) := {v 2 H 1 (⌦)| ⌃ v = 0}.
An important result, which will find widespread application in the sequel, is the so-called Poincaré inequality, see ([25], Chapter 1, Lemma 2.6):

Theorem 2.3 Assume that ⌦ is a bounded connected open set of R N . Then there exists a constant C ⌦ (⌃) > 0 such that Z ⌦ v 2 (x)dx  C ⌦ (⌃) Z ⌦ |rv(x)| 2 dx (2.2.7) for each v 2 H 1 ⌃ (⌦).
We next recall Sobolev embedding theorem ( [START_REF] Robert | Sobolev spaces[END_REF], Chapter 4).

Theorem 2.4 Assume that ⌦ is a (bounded or unbounded) open set of R d with a Lipschitz continuous boundary, and that 1  p < 1. Then the following continuous embedding hold:

1. If 0  sp < d, then W s,p (⌦) ⇢ L p ⇤ (⌦) for p ⇤ = dp/(d sp); 2. If sp = d, then W s,p (⌦) ⇢ L q (⌦)
for any q such that p  q < 1;

3. If sp > d, then W s,p (⌦) ⇢ C 0 ( ⌦).

Variational formulations

A very interesting feature of Darcy's equations is that they admit two equivalent variational formulations, whether the space of velocities is more regular than the space of pressures or not, see [START_REF] Azaïez | The mortar spectral element method in domains of operators. part I: The divergence operator and Darcy's equations[END_REF][START_REF] Achdou | A priori and a posteriori analysis of finite volume discretizations of Darcy's equations[END_REF][START_REF] Bernardi | Mortar finite element discretization for the flow in a nonhomogeneous porous medium[END_REF]. The drag coefficient ↵ will be considered as piecewise constants or piecewise smooth functions. The variational spaces can be choosen as

(u; p) in H(div ; ⌦) ⇥ L 2 (⌦) or in L 2 (⌦) ⇥ H 1 (⌦)
. So two variational formulations will be constructed.

Throughout the thesis, we make the following assumptions on the function ↵: there exists a finite number of domains ⌦ l , 1  l  L, such that :

• they form a partition of ⌦ without overlapping:

⌦ = L X l=1 ⌦l , ⌦ l \ ⌦ l 0 = ;, 1  l 6 = l 0  L, (2.3.1)
•the restriction of ↵ to each ⌦l , 1  l  L, is bounded and positive, i,e. there exists constants ↵ max l and ↵ min l such that

↵ max l = sup x2 ⌦l ↵(x) < +1, ↵ min l = inf x2 ⌦l ↵(x) > 0. (2.3.2)
We set

↵ max = max 1lL ↵ max l , ↵ min = min 1lL ↵ min l .
(2.3.3)

Velocity in H(div , ⌦)

In this subsection, for each domain O in R d with a Lipschitz-continuous boundary, we use the full scale of Sobolev spaces H s (O) and H s 0 (O), s 0, their trace spaces on @O and their dual spaces. We denote by C 1 ( Ō) the space of restrictions to O of indefinitely differentiable functions of R d and by D(O) its subspace of functions with a compact support in O.

Let be any part of @⌦ with positive measure. We also recall that H 1 2 00 ( ) is defined as the space of functions in H 1 2 ( ) such that their extension by zero belongs to H 1 2 (@⌦).

We have the following formula

8q 2 H 1 2 00 ( ), < (v • n), q(x) >= Z ⌦ (div v) q(x)dx + Z ⌦ v • (grad q)(x)dx
where q is any lifting in H 1 (⌦) of the extension by zero of q to @⌦. So the normal trace on of a function v in H(div , ⌦) makes sense in the dual space H

1 2 00 ( ) 0 of H 1 2
00 ( ). In fact, the integral in the left-hand side of the above equality represents a duality pairing.

We next define

H 0 (div , ⌦) = {v 2 H(div , ⌦); v • n = 0 on @⌦} (2.3.4)
Then D(⌦) d is dense in H 0 (div , ⌦), and both H 0 (div , ⌦) and H(div , ⌦) are Hilbert spaces for the scalar product associated with the norm defined in (2.2.1).

We now introduce the variational spaces

X(⌦) = H(div , ⌦), X 0 (⌦) = {v 2 X(⌦); v • n = 0 on 1 }. (2.3.5)
Both of them are equipped with the norm

||v|| X(⌦) = ||v|| H(div ,⌦) = (||v|| 2 L 2 (⌦) d + ||div v|| 2 L 2 (⌦) ) 1 2 . (2.3.6)
The variational problem we consider reads:

Find (u, p) in X(⌦) ⇥ L 2 (⌦) such that u • n = k on 1 and that 8v 2 X 0 (⌦), a ↵ (u, v) + b(v, p) = L(v), 8q 2 L 2 (⌦), b(u, q) = 0, (2.3.7)
where the bilinear forms and linear form are defined by

a ↵ (u, v) = L X l=1 ↵ l Z ⌦ l u(x) • v(x)dx, b(v, q) = Z ⌦ (div v)(x) • q(x)dx. L(v) = L X l=1 ↵ l Z ⌦ l f(x) • v(x)dx Z 2 p 0 (s) (v • n)(s)ds
We make the following assumption on the data

f 2 X 0 (⌦) 0 , k 2 (H 1 2 00 ( 1 )) 0 , p 0 2 H 1 2 00 ( 2 ), (2.3.8) 2.3.2 Velocity in L 2 (⌦)
The variational problem that we consider now reads : Find (u, p) in L 2 (⌦) d ⇥ H 1 (⌦) such that p = p 0 on 2 and that

8v 2 L 2 (⌦) d , a ↵ (u, v) + b(v, p) = L(v), 8q 2 H 1 2 (⌦), b(u, q) = R 1 k(⌧ ) q(⌧ )d⌧, (2.3.9)
where the bilinear forms and linear form are defined by

a ↵ (u, v) = L X l=1 ↵ l Z ⌦ l u(x) • v(x)dx, (2.3.10) b(v, q) = Z ⌦ v(x) • (grad q)(x)dx. (2.3.11)
and Proof: We refer to [START_REF] Bernard | Density results in Sobolev spaces whose elements vanish on a part of the boundary[END_REF].

L(v) = Z ⌦ ↵f(x) • v(x)dx (2.3.12) H 1 2 (⌦) = {q 2 H 1 (⌦), q = 0 on 2 }. ( 2
If Y (⌦) = D( ⌦)( ⌦) \ X 0 (⌦), then Y (⌦) is dense in X 0 (⌦).
When multiplying the first line in (2. To prove the well-posedness of problem (2.3.7), we first construct a lifting of the boundary conditions. Before that we introduce the ↵-dependent norms

||v|| ↵,X(⌦) = ( L X l=1 ↵ l (||v|| 2 L 2 (⌦ l ) d + ||div v|| 2 L 2 (⌦ l ) )) 1 2
(2.4.1)

||q|| ↵ ⇤ ,L 2 (⌦) = ( L X l=1 1 ↵ l ||q|| L 2 (⌦ l ) ) 1 2
(2.4.2)

Lemma 2.2 There exists a divergence-free function u b in X(⌦) which satisfies

u b • n = k on 1 , (2.4.3)
and

||u b || ↵,X(⌦)  C p ↵ max ||k|| (H 1 2 00 ( 1 )) 0 . (2.4.4)
where C is dependent with ↵.

Proof: We now denote by I(⌦) the space

I(⌦) = {µ 2 H 1 (⌦); µ = 0 on 2 },
and we consider the problem :

Find in I(⌦) such that ,

8µ 2 I(⌦), R ⌦ (grad ) • (gradµ)(x)dx = R 1 (k µ)(x)d⌧ (2.4.5)
This problem has a unique solution. Moreover the function u b = grad is divergencefree on ⌦ and satisfies u b • n = k on 1 , and

||u b || ↵,X(⌦)  C p ↵ max ||k|| (H 1 2 00 ( 1 )) 0 .
(2.4.6)

To go further, we set: u 0 = u u b , and consider the following problem: Find (u 0 , p)

in X 0 (⌦) ⇥ L 2 (⌦) such that 8v 2 X 0 (⌦), a ↵ (u 0 , v) + b(v, p) = a ↵ (u b , v) + L(v), 8q 2 L 2 (⌦), b(u 0 , q) = 0. (2.4.7)
Then we can see that the kernel

V (⌦) = {v 2 X 0 (⌦); 8 q 2 L 2 (⌦), b(v, q) = 0} (2.4.8)
coincides with the space of functions in X 0 (⌦) which are divergence-free on ⌦. So it is easy to have the following Lemma about a ↵ (•, •):

Lemma 2.3
The following ellipticity property holds

8v 2 V (⌦), a ↵ (v, v) = ||v|| 2 ↵,X(⌦) .
(2.4.9)

Proof: 8 v 2 V (⌦), a ↵ (v, v) = L P l=1 ↵ l R ⌦ l v • vdx = ||v|| 2 ↵,X(⌦) .
Lemma 2.4 There exists a constant > 0 such that the following inf-sup condition holds

8q 2 L 2 (⌦), sup v2X 0 (⌦) b(v, q) ||v|| ↵,X(⌦) r ↵ min ↵ max ||q|| ↵ ⇤ ,L 2 (⌦) .
(2.4.10)

Proof: We first define the space

W = {x|x 2 H 1 (⌦), x = 0 on 2 },
then for all q 2 L 2 (⌦), their exists Y 2 W , such that

8 > > > < > > > : Y = q, in ⌦ gradY • n = 0, on 1 Y = 0 on 2 (2.4.11) Letting v = grad Y , then v 2 X 0 (⌦) and div v = q in ⌦. Obviously, b(v, q) = Z ⌦ div v q dx = ||q|| 2 L 2 (⌦) .
Thanks to Green's formula and Poincaré-Friedrichs inequality

||grad Y || 2 L 2 (⌦) = Z ⌦ q Y dx  ||q|| L 2 (⌦) ||Y || H 1 (⌦)  c||q|| L 2 (⌦) ||grad Y || L 2 (⌦) .
It follows with

||v|| L 2 (⌦) = ||grad Y || L 2 (⌦)  c||q|| L 2 (⌦) So ||v|| ↵,X(⌦) = ( L X l=1 ↵ l (||v|| 2 L 2 (⌦ l ) d + ||div v|| 2 L 2 (⌦ l ) )) 1 2  C p ↵ max ||q|| L 2 (⌦)
Then we obtain the result

8q 2 L 2 (⌦), sup v2X 0 (⌦) b(v, q) ||v|| ↵,X(⌦) c r ↵ min ↵ max ||q|| ↵ ⇤ ,L 2 (⌦) .
It follows from Lemma 2.3 and Lemma 2.4 that problem (2.4.7) has a unique solution (u 0 , p) in X 0 (⌦) ⇥ L 2 (⌦) and that this solution satisfies

||u 0 || ↵,X(⌦) + c r ↵ min ↵ max ||p|| ↵ ⇤ ,L 2 (⌦)  C 1 p ↵ min (||u b || ↵,X(⌦) + ||f|| L 2 (⌦) + ||p 0 || H 1 2 00 ( 2 )
)

(2.4.12)

We now obtain the main result of this section. ).

( Proof: Due to the density of D(⌦) d in L 2 (⌦) d , the first line in (2.1.1), when taken in the distribution sense, is fully equivalent to the first equation in (2.3.9). On the other hand, when multiplying the second line in (2.1.1) by a function q in D(⌦ [ 1 ), integrating by parts and using the third line, we obtain that the second equation of (2.3.9) is satisfied for all q in C 1 (⌦) \ H 1 2 (⌦), hence for all q in H 1 2 (⌦) by density according to [START_REF] Bernard | Density results in Sobolev spaces whose elements vanish on a part of the boundary[END_REF]. Conversely, letting q run through D(⌦ [ 1 ) gives the second line of (2.1.1) in the distribution sense and letting q run through C 1 (⌦) \ H 1 2 (⌦) leads to the third line.

We now introduce the ↵-dependent norms

||v|| ↵ = ( L X l=1 ↵ l ||v|| 2 L 2 (⌦) d ) 1 2 , ||q|| ↵⇤ = ( L X l=1 1 ↵ l |q| 2 H 1 (⌦ l ) ) 1 2
(2.4.14)

To prove the well-posedness of problem (2.3.9), we first cite the following Property ( [START_REF] Louis | Problemes aux limites non homogenes et applications[END_REF], Chap. I, § 11).

Property 2.1 For all p 0 2 H 1 2 00 ( 2 ), the extension p0 by zero from 2 to @⌦, such that p0

| 2 = p 0 belongs to H 1 2 (@⌦) and satisfies || p0 || H 1 2 (@⌦)  C ||p 0 || H 1 2 00 ( 2 )
.

Then we construct a lifting of the boundary conditions of p0 . We give the following lemma.

Lemma 2.5 There exists a linear continuous map R 0 : H s 1/2 ( ) ! H s (⌦) such that R 0 p0 = P , and inf

0 ( P )= p0 || P || H 1 (⌦) = || p0 || H 1 2 (@⌦)  C||p 0 || H 1 2 00 ( 2 ) (2.4.15)
To go further, we set: P 1 = p P , and consider the following problem: Find

(u, P 1 ) in L 2 (⌦) d ⇥ H 1 2 (⌦) such that 8v 2 L 2 (⌦), a ↵ (u, v) + b(v, P 1 ) = b(v, P ) + L(v), 8q 2 H 1 2 (⌦), b(u, q) =< k, q > 1 .
(2.4.16)

Indeed, the form a ↵ (•, •) and b(•, •) satisfy the continuity properties and we have the ellipticity property

8 v 2 L 2 (⌦) d , a ↵ (v, v) = ||v|| 2 ↵ .
(2.4.17)

We also have the inf-sup condition on the form b(•, •) as follows:

Lemma 2.6 8 q 2 H 1 2 (⌦), sup v2L 2 (⌦) d b(v, q) ||v|| ↵ ||q|| ↵⇤ (2.4.18) Proof: 8 q 2 H 1 2 (⌦), letting v = ↵ 1 rq, then v 2 L 2 (⌦) d . b(v, q) = R ⌦ v • rq = ||q|| 2 ↵ ⇤ So b(v,q) ||v||↵ ||q|| 2 ↵ ⇤ ||q|| ↵ ⇤ ||q|| ↵ ⇤ It
Find (u 0 , P 1 ) in L 2 (⌦) d ⇥ H 1 2 (⌦) such that 8v 2 L 2 (⌦), a ↵ (u 0 , v) + b(v, P 1 ) = b(v, P ) a ↵ (u b , v) + L(v), 8q 2 H 1 2 (⌦), b(u 0 , q) = 0.
(2.4.23) has a unique solution (u 0 , P 1 ) which morever satisfies

||u 0 || ↵ + ||P 1 || ↵⇤  C(||u b || ↵ + ||f|| ↵ + || P || ↵ ⇤ ) (2.4.24)
Then, the pair (u, p) with u = u 0 + u b , p = P

Regularity properties

Let O be a subdomain of ⌦ such that ↵ is a constant in a neighbourhood of Ō in ⌦. We first set

H(curl, div ; O) = H(curl, O) \ H(div , O).
Then we introduce a space

W 0 = {u 2 H(curl, div ; O)|u • n 2 L 2 (@O)}.
Due to ([5], Remark 2.16) and [START_REF] Costabel | Un résultat de densité pour les équations de maxwell régularisées dans un domaine lipschitzien[END_REF], when O is a general Lipschitz domain, the space

W 0 is imbedded in H 1 2 (
H s (⌦) d ⇥ H s 1 2 (⌦) ⇥ H s+ 1 2 (⌦) into H s (O) d ⇥ H s+1 (O), (i) for 0  s  1 2 in the general case, (ii) for 0  s  1 when Ō is convex.
It follows from [START_REF] Costabel | Singularities of Maxwell interface problems[END_REF] that these results still hold with O replaced by O \ ⌦ l i) in dimension d = 2, when the ⌦ l are polygons and Ō does not contain any vertex of the ⌦ l , ii) in dimension d = 3, when the ⌦ l are polyhedras and Ō neither contains a vertex of the ⌦ l nor intersects an edge of the ⌦ l .

However when Ō either contains a vertex of the ⌦ l or intersects an edge of the ⌦ l and when we consider the global regularity properties, the results will be weaker. More properties can be proved only in dimension d = 2 and thus we only consider this case in what follows.

We first consider the local regularity properties. We introduce here also a space

W = {u 2 H(curl, div ; ⌦)|u • n 2 L 2 (@⌦)}.
We know from [START_REF] Costabel | Un résultat de densité pour les équations de maxwell régularisées dans un domaine lipschitzien[END_REF] when ⌦ is a polygon, a function u in W can be written as

u = u r + gradS, (2.5.1)
where u r belongs to H 1 (⌦) 2 and S is a linear combination of the singularities of the Laplace equation provided with Neumann boundary conditions. We recall that each singularity in the neighbourhood of a corner of the polygon with aperture ! has the form

r ⇡ ! ('(✓) + (logr) p (✓)),
where r is the distance to the corner, ✓ the corresponding angular variable, p is equal to 0 except when ⇡ ! is an integer where it is equal to 1. More generally, any such function u which has the further property

div u 2 H s (⌦), curl u 2 H s (⌦) 3 , (2.5.2)
admits the expansion (2.5.1) with u r in H s+1 (⌦) 2 for all s, 0 < s < 2⇡ ! 1.

Taking the curl of the fisrt equation in (2.1.1), which yields

curl (↵u) = curl (↵f) in ⌦.
So when ⌦ is a polygon, u admits the expansion (2.5.1), and if curl f belongs to

H s (⌦) 2 , 0 < s < 2⇡ ! 1
, where ! denotes the largest angle of ⌦, the regular part u r in this expansion belongs to H s+1 (⌦).

We next give the regularity property of p. Taking div of the first equation in (2.1.1), we consider the following mixed boundary conditions problem:

8 > > > < > > > : div ( 1 ↵ grad p) = div f in ⌦, 1 ↵ @ n p = f • n on , p = p 0 on 2 (2.5.3)
We know from [START_REF] Dauge | Neumann and mixed problems on curvilinear polyhedra[END_REF] that there exists a real number s 0 , 0 < s 0 < 1 2 , only depending on the geometry of ⌦, ↵ min and ↵ max , such that when div f 2 H s 1 (⌦) and p 0 2

H s+ 1 2 ⌃
( 2 ), the solution p of the equation (2.5.3) belongs to H s+1 (⌦) for all s  s 0 , where H

s+ 1 2 ⌃
( 2 ) stands for the space of functions such that their restrictions to any edge e contained in ¯ 2 belongs to H s+ 1 2 (e). Morever, when ⌦ is a polygon, according to ( [START_REF] Grisvard | Boundary Value Problems in Non-Smooth Domains[END_REF],Chapter 5) and [START_REF] Dauge | Neumann and mixed problems on curvilinear polyhedra[END_REF], p will belong to

H s+1 (⌦), s  min{s 0 , 2⇡ ! 1}.
We then consider the local regularity properties when Ō contains either vertex of the ⌦ l or intersects an edge of the ⌦ l . we are now at the last conclusion of this section.

Suppose f| ⌦ l , k, p 0 belongs to H s l (⌦ l ) 2 ⇥ H s l 1 2 (⌦ l ) ⇥ H s l + 1 2 (⌦ l )
for any real positive number s l : i) When ⌦l contains no corner of ⌦, the regularity of (u, p) on ⌦ l only depends on the data f, and (u, p) belongs to

H s l (⌦ l ) 2 ⇥ H s l +1 (⌦ l ).
ii) When ⌦l contains a corner of ⌦ with !, we derive from (2.5.1) that u| ⌦ l = u r + grad S, p| ⌦ l = p r + S. So the first equation of (5.1.1) can be rewritten as

1 ↵ grad p 0 = f u r wherep 0 = p r + (1 + ↵) S. (2.5.4)
Taking the curl of the first equation of (5.1.1) and taking div of the equation (2.5.4), we can get that (u r , p r ) belongs to

H s (⌦ l ) 2 ⇥ H s+1 (⌦ l ) for s  s l and s < 2⇡ ! .
Chapter 3

Spectral Discretization of Darcy's Equations

Orthogonal polynomials

We firstly work on the interval ⇤ = [START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF][START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF], since all results on ⌦ are deduced from their analogues on ⇤ thanks to tensorization arguments. Over ⇤, all the approximation properties rely on a proper choice of an orthogonal basis of L 2 (⇤) : the Legendre polynomials.

The family of Legendre polynomials is a family (L k ) k 0 of polynomials in one variable, which are orthogonal to each other in L 2 (⇤). For any integer k 0, the polynomail L k has degree k. If L k (x) is normalized so that L k (1) = 1, then for any k:

L k (x) = 1 2 k [k/2] X (l=0) ( 1) l ✓ k l ◆✓ 2k 2l k ◆ x k 2l , (3.1.1)
where

[k/2] denotes the integral part of k/2.
It can be noted that, for all k, the k zeros of L k are distinct real numbers in ⇤ and also that the polynomial L k is even if k is even and odd if k is odd.

The norm of each L k is given by the formula

Z 1 1 L 2 k (x)dx = 1 k + 1 2 . (3.1.2)
We recall from ([1], Chapter 22) some very important properties of these polynomials, firstly the differential equation

((1 x 2 )L 0 k ) 0 + k(k + 1)L k = 0, k 0, (3.1.3)
secondly the integral equation

(2k + 1)L k = L 0 k+1 L 0 k 1 , k 1, (3.1.4)
thirdly the induction formula, which allows for computing them:

L k+1 (x) = 2k + 1 k + 1 xL k (x) k k + 1 L k 1 (x), (3.1.5)
where L 0 (x) = 1 and L 1 (x) = x.

We consider now discrete Legendre series. Since explicit formulas for the quadrature nodes are not known, such points have to be computed numerically as zeros of approximate polynomials. The quadrature weights can be expressed in closed form in terms of the nodes, as indicated in the following formulas: ([44], Chapter 2)

• Legendre-Gauss (LG) : forj = 1, . . . , N

x j are zeros of L N +1 ;

(3.1.6)

! j = 2 (1 x 2 j )[L 0 N +1 (x j )] 2 .
(3.1.7)

• Legendre -Gauss-Lobatto (LGL) : forj = 0, . . . , N

x j are zeros of (1

x 2 )L 0 N ; (3.1.8) ! j = 2 N (N + 1) 1 [L N (x j )] 2 .
(3.1.9)

Quadrature Formula

As we know that, the most natural quadrature formula in the polynomial context is the Gauss Lobatto formula associated with the Lebesgue measure, since it allows for easily handling boundary conditions and leads to diagonal mass matrices. We firstly describe it, next we study the corresponding interpolation operators.

The Gauss Lobatto formula Let us set x 0 = 1 and x N = 1. There exists a unique set of N 1 nodes x j , 1  j  N 1 and a unique set of N + 1 weights ⇢ j , 0  j  N , such that the following exactness property holds

8 2 P 2N 1 (⇤), Z 1 1 (x)dx = N X j=0 (x j )⇢ j . (3.2.1)
The zeros x j , 1  j  N 1, are the zeros of L 0 N and the weights ⇢ j are positive, given by

⇢ j = 2 N (N + 1)L 2 N (x j ) , 0  j  N. (3.2.2)
We conclude with a further useful property of the quadrature formula.

Lemma 3.1 There exists a constant c such that

8 N 2 P N (⇤), || N || 2 L 2 (⇤)  N X j=0 2 N (x j )⇢ j  3|| N || 2 L 2 (⇤) . ( 3

.2.3)

Proof: We write the expansion of any polynomial N in L 2 (⇤) :

N = N X n=0 ↵ n L n . (3.2.4)
We deduce from the exactness property (3.2.1) that

|| N || 2 L 2 (⇤) = N X n=0 ↵ 2 n ||L n || 2 L 2 (⇤) , N X j=0 2 N (x j )⇢ j = N 1 X n=0 ↵ 2 n ||L n || 2 L 2 (⇤) + ↵ 2 N N X j=0 L 2 N (x j )⇢ j ,
so that it suffices to prove the results with N = L N . Noting that the degree of the

polynomial L 2 N + N 2 (1 x 2 )L 0 2
N is less than 2N 1 (this comes by computing the coefficient of x 2N ), we derive that

Z 1 1 (L 2 N (x) + N 2 (1 x 2 )L 0 2 N (x))dx = N X j=0 (L 2 N (x j ) + N 2 (1 x 2 j )L 0 2 N (x j ))⇢ j , (3.2.5)
or equivalently, since the x j are the zeros of (1

x 2 )L 0 N , N X j=0 L 2 N (x j )⇢ j = Z 1 1 L 2 N (x)dx + N 2 Z 1 1 L 0 2 N (x)(1 x 2 )dx. (3.2.6)
The second integral is easily computed, leading to

N X j=0 L 2 N (x j )⇢ j = (2 + 1 N )||L N || 2 L 2 (⇤) . (3.2.7)
Since N 1, this gives the desired result.

Polynomial approximation

Polynomial approximation on the interval

Let now N be a fixed positive integer 2. In what follow, c stands for a positive constant independent of N . We introduce the orthogonal projection operator ⇡ N from L 2 (⇤) onto P N (⇤). Then the projection operator can be expressed in the following way:

8 2 L 2 (⇤), ⇡ N = N X n=0 n L n , (3.3.1)
where

n = 1 ||L n || 2 L 2 (⇤) Z 1 1 (x)L n (x)dx. (3.3.2)
Lemma 3.2 For any real numbers s 0, the following estimate holds for any in H s (⇤):

|| ⇡ N || L 2 (⇤)  cN s || || H s (⇤) . (3.3.3)
For the proof of this lemma, we propose ( [START_REF] Bernardi | Spectral, spectral element and mortar element methods[END_REF], Proposition 1).

We now consider the orthogonal projection operator ⇡ 1,0 N from H 1 0 (⇤) onto P 0 N (⇤), for the scalar product associated with the norm

| • | H 1 (⇤) . Its properties are derived from the formula (⇡ 1,0 N ) 0 = ⇡ N 1 0 for the bounded in H 1 (⇤),
(⇤) \ H 1 0 (⇤): | ⇡ 1,0 N | H 1 (⇤) + N || ⇡ 1,0 N || L 2 (⇤)  cN 1 s || || H s (⇤) . (3.3.4)
Finally, with any function in H 1 (⇤), hence continuous, we associate the function 0 defined by

0 (x) = (x) ( 1) 1 x 2 (1) 1 + x 2 , (3.3.5)
and since it belongs to H 1 0 (⇤), we define the polynomial ⇡ 1 N by

(⇡ 1 N )(x) = (⇡ 1,0 N 0 )(x) + ( 1) 1 x 2 + (1) 1 + x 2 . (3.3.6)
This operator does not correspond to an orthogonal projection, but it preserves the values in ±1 and its properties are easily derived from the formula

⇡ 1 N = 0 ⇡ 1,0 N 0 . Proposition 3.2 ([23],
Proposition 3) For any real number s 1, the following estimate holds for any in

H s (⇤) \ H 1 0 (⇤): | ⇡ 1 N | H 1 (⇤) + N || ⇡ 1 N || L 2 (⇤)  cN 1 s || || H s (⇤) . (3.3.7)

Polynomial approximation on the square or cube

For each n 0, let P n (⌦) be the space of polynomials worth degree  n with respect to each variable, and P 0 n be its subspace made of all polynomials which vanish on the boundary @⌦.

As previously, we fix an integer N 2. The best fit of a given function is realized via projection operators onto P N (⌦) or P 0 N (⌦), and analyzing their properties relies on tensorization arguments. For instance, if ⇧ N denotes the orthogonal projection

operator from L 2 (⌦) onto P N (⌦), it coincides with ⇡ (x) N ⇡ (y) N in dimension d = 2, with ⇡ (x) N ⇡ (y) N ⇡ (z)
N in dimension d = 3, where the exponent after an operator denotes the direction in which it is applied. So the approximation properties of the operator ⇧ N are derived from the triangular inequality, in dimension d = 2 for instance: [START_REF] Bernardi | Spectral, spectral element and mortar element methods[END_REF], Proposition 4) For any real number s 0, the following estimate holds for any v in H s (⌦) :

||v ⇧ N v|| L 2 (⌦)  ||v ⇡ (x) N v|| L 2 (⌦) + ||⇡ (x) N (v ⇡ (y) N v)|| L 2 (⌦) . (3.3.8) Proposition 3.3 ([
||v ⇧ N v|| L 2 (⌦)  cN s ||v|| H s (⌦) . (3.3.9)
Let now ⇧ 1,0 N stand for the orthogonal projection operator from H 1 0 (⌦) onto P 0 N (⌦). Using a further duality argument, we derive the next proposition.

v in H s (⌦) \ H 1 0 (⌦) : |v ⇧ 1,0 N v| H 1 (⌦) + N ||v ⇧ 1,0 N v|| L 2 (⌦)  cN 1 s ||v|| H s (⌦) . (3.3.10)
Finally, the approximation properties of the orthogonal projection operator ⇧ 1 N from H 1 (⌦) onto P N (⌦) are derived from the same triangular inequality .

Proposition 3.5 ([23], Proposition 6) For any real number s 1, the following estimate holds for any v in H s (⌦) :

|v ⇧ 1 N v| H 1 (⌦) + N ||v ⇧ 1 N v|| L 2 (⌦)  cN 1 s ||v|| H s (⌦) . (3.3.11)

Polynomial interpolation

In this section, we will introduce the interpolation operator on Gauss-Lobatto points.

Polynomial interpolation on the interval

Our aim is to derive the appxoximation properties of the Lagrange interpolation operator i N at the nodes x j , 0  j  N , with values in P N (⇤), which means that

(i N )(x j ) = (x j ), 0  j  N. (3.4.1)

Lemma 3.3 ([23],

Lemma 3) There exists a positive constant c such that, for any function in

H 1 0 (⇤), ||i N || L 2 (⇤)  c(|| || L 2 (⇤) + N 1 | | H 1 (⇤) ). (3.4.2)
Applying the above inequality to the function ⇡ 1 N obviously gives an estimate for the interpolation error of functions which belongs to H 1 (⇤). However, we have rather state a slightly more general result which allows for interpolating less smooth functions.

Proposition 3.6 ([23], Proposition 7) For any real number s > 1 2 , the following estimate holds for any in H s (⇤): [START_REF] Bernardi | Spectral methods[END_REF], Theorem 13.4) For any real number r and s satisfying s > max{ 1 2 , r} and 0  r  1, there exists a postive constant c depending only on s such that, for any function in H s (⇤), the following estimate holds:

|| i N || L 2 (⇤)  cN s || || H s (⇤) . (3.4.3) Proposition 3.7 ([
| i N | H r (⇤)  cN r s || || H s (⇤) . (3.4.4)

Polynomial interpolation on the square or cube

By tensorization, we associate with the nodes x j the grid

⌅ N = 8 < : {(x i , x j ), 0  i, j  N } in dimension d=2, {(x i , x j , x k ), 0  i, j, k  N } in dimension d=3, (3.4.5)
and also the discrete product: for all continuous functions u and v on ⌦,

((u, v)) N = 8 > > < > > : N P i=0 N P j=0 u(x i , x j )v(x i , x j )⇢ i ⇢ j in dimension d=2, N P i=0 N P j=0 N P k=0 u(x i , x j , x k )v(x i , x j , x K )⇢ i ⇢ j ⇢ k in dimension d=3, ( 3
||v I N v|| L 2 (⌦)  cN s ||v|| H s (⌦) .
(3.4.7)

For any real number r and s satisfying s > max{ d 2 , r} and 0  r  1, there exists a postive constant c depending only on s such that, for any function in H s (⌦), the following estimate holds:

|v I N v| H r (⌦)  cN r s ||v|| H s (⌦) .
(3.4.8)

Spectral discretization of the Darcy's problem

According to the variational forms in Chapter 2, we here discuss the discrete problems.

First discretization

According to the variational formulation, we introduce the discrete spaces as follows:

D N = P N (⌦) d , (3.5.1)
also we introduce another space:

D 0 N = {u N 2 D N , u N • n = 0 on 1 }. (3.5.2)
For the space of pressure, there are different choices. Usually we consider the spurious modes on the pressure, namely the set

Z n = {q N 2 P N (⌦), 8v N 2 D 0 N , b N (v N , q N ) = 0} (3.5.3)
which is analyzed in [START_REF] Azaïez | Spectral methods applied to porous media[END_REF] and [START_REF] Bernardi | Discrétisations variationnelles de problemes aux limites elliptiques[END_REF]. The space of pressure M N will be taken as the orthogonal space of Z n in P N .

But here we choose another pressure space as M N = P N 2 (⌦) which has not been discussed yet in the literature as far as the author's knowledge. Assuming that the functions f, k, p 0 has continuous restrictions to ⌦ and ¯ 1 , ¯ 2 respectively. Then the discrete problem built from the variational formulation reads: Find (u N , p N ) in

D N ⇥ M N such that u N • n = I 1 N k on 1 , (3.5.4)
and

8v N 2 D 0 N , a N ↵ (u N , v N ) + b N (v N , p N ) = L N (v N ), 8q N 2 M N , b N (u N , q N ) = 0. (3.5.5)
where the bilinear forms a N ↵ (•, •) and b N (•, •) are defined by

a N ↵ (u N , v N ) = ↵((u N , v N )) N , b N (v N , q N ) = ((div v N , q N )) N , L N (v N ) = ((↵ f, v N )) N ((p 0 , v N • n)) 2
N .

(3.5.6)

Next we introduce the ↵ dependent norms

||v N || ↵ = (↵(||v N || 2 + ||div v N || 2 )) 1 2 , ||q N || ↵ ⇤ = (↵ 1 ||q N || 2
u b N • n = I 1 N k on 1 , (3.5.8)
and

||u b N || ↵  C p ↵ max ||I 1 N k|| (H 1 2 00 ( 1 )) 0 . (3.5.9)
where C is dependent with ↵.

To go further, we set: u 0 N = u u b N , and consider the following problem: Find

(u 0 N , p) in D 0 N ⇥ M N such that 8v N 2 D 0 N , a N ↵ (u 0 N , v) + b(v, p) = a N ↵ (u b N , v) + L N (v N ), 8q 2 M N , b N (u 0 N , q) = 0.
(3.5.10)

Then we can see that the kernel

V N = {v N 2 D 0 N ; 8 q N 2 M N , b N (v N , q N ) = 0} (3.5.11)
coincides with the space of functions in D 0 N which are divergence-free on ⌦. So it is easy to have the following Lemma about a N ↵ (•, •):

Lemma 3.5 There exists a constant c independent of the discretization, such that the following ellipticity property holds

8v N 2 V N , a N ↵ (v N , v N ) c||v N || 2 ↵ .
(3.5.12)

Next, we come to the inf-sup condition.

Lemma 3.6 There exists a constant > 0 such that the following inf-sup condition holds

8q N 2 M N , sup v N 2D 0 N b N (v N , q N ) ||v N || ↵ ||q N || ↵ ⇤ . (3.5.13)
Proof: The proof is given only in the case d = 2, since the corresponding proof in the case d = 3 is completely similar. Any function q N in M N has the expansion

q N (x, y) = N 2 X m=0 N 2 X n=0 ↵ mn L m (x)L n (y). (3.5.14)
We can derive that

||q N || 2 L 2 (⌦) = N 2 X m=0 N 2 X n=0 ↵ 2 mn 1 (m + 1 2 )(n + 1 2 ) (3.5.15)
The idea is now to choose v N = (v1 N , v2 N ), with

v1 N (x, y) = N 2 X m=0 m X n=0 ↵ mn L m+1 (x) L m 1 (x) 2m + 1 L n (y) (3.5.16)
and

v2 N (x, y) = N 2 X m=0 N 2 X n=m+1 ↵ mn L m (x) L m+1 (y) L m 1 (y) 2m + 1 . (3.5.17)
In fact, the polynomial v N belongs to D 0 N , and which satisfies div v N = q N . Next, we note that

@v1 N (x, y) @x = N 2 X m=0 m X n=0 ↵ mn L m (x)L n (y) (3.5.18)
which allows for bounding || @v1 N (x,y)

@x
|| L 2 (⌦) by comparing with (3.5.15). Using a similar argument for @v2 N (x,y) @y , we derive that

|| @v1 N (x, y) @x || L 2 (⌦) + || @v2 N (x, y) @y || L 2 (⌦)  c||q N || L 2 (⌦) , (3.5.19)
and also, thanks to the Poincaré-Friedrichs inequality applied with respect either to

x or y,

||v1 N (x, y)|| L 2 (⌦) + ||v2 N (x, y)|| L 2 (⌦)  c||q N || L 2 (⌦) , (3.5.20)
then the inf-sup condition is obviously.

It follows from Lemma 3.5.12 and Lemma 3.6 that problem (3.5.10) has a unique solution (u 0 N , p) in D 0 N ⇥ M N and that this solution satisfies

||u 0 || ↵ + ||p|| ↵ ⇤  C(||u b || ↵ + ||I N f|| L 2 (⌦) d + ||I 2 N p 0 || H 1 2 00 ( 2 ) ) (3.5.21)
We now state the main result of this section. Moreover this solution satisfies

||u N || ↵ + ||p N || ↵ ⇤  c(||I 1 N k|| (H 1 2 ) 0 00 ( 1 ) + ||I N f|| L 2 (⌦) d + ||I 2 N p 0 || H 1 2 00 ( 2 )
). • Error estimate : here we introduce a new norm as follows:

||u N || ↵,L 2 (⌦) = (↵||u N || 2 ) 1 2 .
We prove an error estimates, first for the velocity, second for the pressure. Let u be the solution of equation ( 2

↵ Z ⌦ u 0 • ! ! ! N dx + b(! ! ! N , p) = ↵ Z ⌦ f • ! ! ! N dx ↵ Z ⌦ u b • ! ! ! N dx. (3.5.23)
Using the definition of V N (⌦) thus implies, for any q N in M N (⌦),

↵ Z ⌦ u 0 • ! ! ! N dx + b(! ! ! N , p q N ) = ↵ Z ⌦ f • ! ! ! N dx ↵ Z ⌦ u b • ! ! ! N dx. ( 3
||u 0 N v N || 2 ↵,L 2 ⌦  ↵ Z ⌦ (u 0 v N ) • (u 0 N v N )dx + ↵ Z ⌦ v N • (u 0 N v N )dx ↵((v N , u 0 N v N )) k N + b(u 0 N v N , p q N ) + ↵((f, u 0 N v N )) k N ↵ Z ⌦ f • (u 0 N v N )dx + ↵ Z ⌦ u b • (u 0 N v N )dx ↵((u b N , u 0 N v N )) k N + Z 2 p 0 • [(u 0 N v N ) • n]ds ((p 0 , (u 0 N v N ) • n)) 2 N .
(3.5.25)

Using a triangle inequality yields

||u u N || ↵,L 2 (⌦)  ||u 0 v N || ↵,L 2 (⌦) + ||u 0 N v N || ↵,L 2 (⌦) + ||u b u b N || ↵,L 2 (⌦)  C{ inf v N 2V N ||u 0 v N || ↵,L 2 (⌦) + sup ! ! ! N 2D N { ↵ R ⌦ v N • ! ! ! N dx ↵((v N , ! ! ! N )) k N ||! ! ! N || ↵,L 2 (⌦) + ↵((f, ! ! ! N )) k N ↵ R ⌦ f • ! ! ! N dx ||! ! ! N || ↵,L 2 (⌦) + ↵ R ⌦ u b • ! ! ! N dx ↵((u b N , ! ! ! N )) k N ||! ! ! N || ↵,L 2 (⌦) + R 2 p 0 • (! ! ! N • n)ds ((p 0 , (! ! ! N • n)) 2 N ||! ! ! N || ↵,L 2 (⌦) + b(! ! ! N , p q N ) ||! ! ! N || ↵,L 2 (⌦) + ||u b u b N || ↵,L 2 (⌦) }.
(3.5.26)

All the items above can be found in ( [START_REF] Bernardi | Discrétisations variationnelles de problemes aux limites elliptiques[END_REF], Chapter V), we will not prove here and we will give the first theorem about the velocity as follows.

||u u N || ↵,L 2 (⌦)  c(N m (||u|| H m (⌦) d + ||p|| H m (⌦) ) +N r ||f|| H r (⌦) d + N s ||k|| H s ( 1 ) + N t ||p 0 || H t ( 2 ) ). (3.5.27)
Due to the inf sup condition, for any q N in M N (⌦), we derive that

D ||p N q N || ↵ ⇤  sup v N 2D 0 N (⌦) R ⌦ (p N q N )(div v N )dx ||v N || ↵ . (3.5.28)
In order to evaluate R ⌦ (p N q N )(div v N )dx, we first use the discrete problem (3.5.5):

Z ⌦ (p N q N )(div v N )dx = ↵((f, v N )) k N ((p 0 , v N • n)) 2 N ↵((u N , v N )) k N + Z ⌦ q N (div v N )dx.
Z ⌦ (p N q N )(div v N )dx = ↵((f, v N )) k N ↵ Z ⌦ f • v N dx +↵ Z ⌦ u • v N dx ↵((u, v N )) k N + Z 2 p 0 • (v • n) ds ((p 0 , v N • n)) 2 N + Z ⌦ (q N p) divv N dx (3.5.30)
Now, it is easy to derive the next theorem.

||u u N || ↵,L 2 (⌦) + ||p p N || ↵ ⇤  c(N m (||u|| H m (⌦) d + ||p|| H m (⌦) ) +N r ||f|| H r (⌦) d + N s ||k|| H s ( 1 ) + N t ||p 0 || H t ( 2 ) ).
(3.5.31)

Second discretization

• Well-posedeness of the solution : According to the variational formulation, we introduce the discrete spaces as follows:

X N = P N (⌦) d , M N = P N (⌦), (3.5.32)
also we introduce another space:

M 1 N = {q N 2 M N , q N = 0 on 2 }. (3.5.33)
Assuming that the functions f, k, p 0 has continuous restrictions to all ⌦k , 1  k  K and ¯ 1 , ¯ 2 respectively. Then the discrete problem built from the variational formulation reads:

Find (u N , p N ) in X N ⇥ M N such that p N = p 0 on 2 , (3.5.34) and 8v N 2 X N , a N ↵ (u N , v N ) + b N (v N , p N ) = L N (v N ), 8q N 2 M 1 N , b N (u N , q N ) = ((k, q N )) 1 N . (3.5.35)
where the bilinear forms a N ↵ (•, •) and b N (•, •) are defined by

a N ↵ (u N , v N ) = ↵((u N , v N )) N , b N (v N , q N ) = ((v N , grad q N )) N , L N (v N ) = ((↵ f, v N )) N .
(3.5.36)

We introduce here related norm and semi-norm as follows:

||u N || ↵ = (↵||u N || 2 ) 1 2 , ||q N || ↵ ⇤ = (↵ 1 |q N | 2 ) 1 2 .
We construct a lifting of the boundary condition of p 0 , we give the following lemma according to the reference ( [START_REF] Bernardi | Discrétisations variationnelles de problemes aux limites elliptiques[END_REF], Th. III. and

||p b N || ↵ ⇤  c r 1 ↵ min ||I 2 N p 0 || H 1 2 ( 2 ) . (3.5.38)
To go further, we set p 0 N = p N p b N , and consider the following problem: Find

(u N , p 0 N ) in X N ⇥ M 1 N such that 8v N 2 X N , a N ↵ (u N , v N ) + b N (v N , p 0 N ) = L N (v N ) b N (v N , p b N ), 8q N 2 M 1 N , b N (u N , q N ) = ((k, q N )) 1 N .
(3.5.39)

Lemma 3.8 The form a N ↵ (•, •) satisfies the following continuity and ellipticity properties

8u N 2 X N , 8v N 2 X N , a N ↵ (u N , v N )  c||u N || ↵ ||v N || ↵ , (3.5.40)
8u N 2 X N , a N ↵ (u N , u N ) ||u N || 2 ↵ . ( 3
((u, v)) N  ((u, u)) 1 2 N ((v, v)) 1 2
N .

(3.5.42)

For all u N in X N , because ↵ k is piecewise constant, we have

a N ↵ (u N , u N ) = ↵ N X i=0 N X j=0 u 2 N (x i , x j )⇢ i ⇢ j , (3.5.43)
then we can easily derive:

8u N 2 X N , ||u N || 2 ↵  a N ↵ (u N , u N )  c||u N || 2 ↵ . (3.5.44)
We next come to the inf-sup condition:

Lemma 3.9 The bilinear form b N (•, •) satisfies the inf-sup condition

8q N 2 M 1 N , sup v N 2X N b N (v N , q N ) ||v N || ↵ C ||q N || ↵ ⇤ . (3.5.45)
Proof: We note that for any q N in M 1 N , we can define the function v N by

v N = ↵ 1 gradq N , (3.5
||u N || ↵ + ||p 0 N || ↵ ⇤  C (||p b N || ↵ ⇤ + ||I 1 N k|| H 1 2 ( 1 ) + ||I N f|| L 2 (⌦) d ) (3.5.47)
Now we are in the position of the main result of this section:

||u N || ↵ + ||p N || ↵ ⇤  C (||I 2 N p 0 || H 1 2 ( 2 ) + ||I 1 N k|| H 1 2 ( 1 ) + ||I N f|| L 2 (⌦) d ) (3.5.

48)

Proof: We establish successively the existence and uniqueness of the solution.

1) It follows from the Lax-milgram lemma, combined with Bramble-Hilbert inequality, that there exists a unique ' N in X N such that

8 N 2 M 1 N , ((grad ' N , grad N )) N = ((k, N )) 1 N . ( 3
(u 0 N , p 0 N ) in X N ⇥ M 1 N such that 8v N 2 X N , a N ↵ (u 0 N , v N ) + b N (v N , p 0 N ) = L N (v N ) b N (v N , p b N ) a N ↵ (u b N , v N ), 8q N 2 M 1 N , b N (u N , q N ) = 0. (3.5.51)
has a unique solution (u 0 N , p 0 N ) which satisfies

||u 0 N || ↵ + ||p 0 N || ↵ ⇤  C (||p b N || ↵ ⇤ + ||u b N || ↵ + ||I N f|| L 2 (⌦) d ) (3.5.52)
Then, the pair To conclude, we introduce the discrete kernal

(u N , p N ), with u N = u 0 N + u b N , p N = p 0 N + p b N ,
V N = {v N 2 X N ; 8q N 2 M 1 N , b N (v N , q N ) = 0}. (3.5.53)
As usual, it plays a key role in the numerical analysis of problem (3.5.34) -(3.5.35).

• Error estimate

This section is devoted to the error estimates, first for the velocity, second for the pressure. We intend to prove here an estimate between the solution (u, p) of problem Let ! ! ! N be any function in the kernal V N . Multiplying the first line of (2.1.1) by

! ! ! N gives a ↵ (u 0 , ! ! ! N ) + b(! ! ! N , P 1 ) = ↵ Z ⌦ f • ! ! ! N dx a ↵ (u b , ! ! ! N ) b(! ! ! N , P ). (3.5.54)
which implies, due to the defination of V N , that for any

q N in M 1 N (⌦), a ↵ (u 0 , ! ! ! N ) + b(! ! ! N , P 1 q N ) = ↵ Z ⌦ f • ! ! ! N dx a ↵ (u b , ! ! ! N ) b(! ! ! N , P ). (3.5.55)
Next, we deduce from the ellipticity property, that we have for any

v N in V N ||u 0 N v N || 2 ↵  a N ↵ (u 0 N v N , u 0 N v N ). ( 3
||u 0 N v N || 2 ↵  a N ↵ (u 0 N v N , u 0 N v N )  a ↵ (u 0 v N , u 0 N v N ) + a ↵ (v N , u 0 N v N ) a N ↵ (v N , u 0 N v N ) + b(u 0 N v N , P 1 q N ) + ((↵f, u 0 N v N )) N ↵ Z ⌦ f • (u 0 N v N )dx + a ↵ (u b , u 0 N v N ) a N ↵ (u b , u 0 N v N ) + b(u 0 N v N , P ) b N (u 0 N v N , p b N ). (3.5.57) It is obvious that 8v N 2 X N , 8q N 2 M 1 N , b N (v N , q N )  ||v N || ↵ ||q N || ↵ ⇤ . (3.5.58)
So, combining the above inequality and triangle inequality, we derive that the error ||u u N || ↵ is bounded, up to a multiplicative constant, by the sum of seven terms:

• the approximation error in

X N inf v N 2V N ||u 0 v N || ↵ , (3.5.59)
• the error approximation in M 1

N inf q N 2M 1 N ||P 1 q N || ↵ ⇤ , (3.5.60)
• three terms issued from numerical integration

sup ! ! ! N 2X N a ↵ (v N , ! ! ! N ) a N ↵ (v N , ! ! ! N ) ||! ! ! N || ↵ , (3.5.61) sup ! ! ! N 2X N a ↵ (u b , ! ! ! N ) a N ↵ (u b N , ! ! ! N ) ||! ! ! N || ↵ , (3.5.62)
sup We refer ([32], Section 2.6.2), the following estimate could be found there.

! ! ! N 2X N ((↵f, ! ! ! N )) ↵ ↵ R ⌦ f • ! ! ! N dx ||! ! ! N || ↵ , (3.5.63) sup ! ! ! N 2X N b(! ! ! N , P ) b N (! ! ! N , p b N ) ||! ! ! N || ↵ . ( 3
For r 2, sup

! ! ! N 2X N ((↵f, ! ! ! N )) ↵ ↵ R ⌦ f • ! ! ! N dx ||! ! ! N || ↵  cN r ||f|| H r (⌦) d . (3.5.66) For s 1, sup ! ! ! N 2X N a ↵ (u b , ! ! ! N ) a N ↵ (u b N , ! ! ! N ) ||! ! ! N || ↵  cN s ||u b || H s (⌦) d . ( 3
inf v N 2V N ||u 0 v N || ↵  cN m ||u 0 || H m (⌦) d . (3.5.68) Then, inf q N 2M 1 N ||P 1 q N || ↵ ⇤  cN m ||P 1 || H m+1 (⌦) . ( 3
||u u N || ↵  c(N m (||u|| H m (⌦) d + ||p|| H m+1 (⌦) ) +N r ||f|| H r (⌦) d + N s ||k|| H s ( 1 ) + N t ||p 0 || H t ( 2 )
).

(3.5.70) Then we come to the error estimate for the pressure. From the inf sup condition (3.5.45), we derive that, for any q N in M 1 N ,

||p 0 N q 0 N || ↵ ⇤  sup v N 2X N b N (v N , p 0 N q 0 N ) ||v N || ↵ . ( 3
b N (v N , p 0 N q 0 N ) = ((↵ f, v N )) N a N ↵ (u N v N ) b N (v N , p b N ) b N (v N , q 0 N ). (3.5.72)
Next, we apply equation (2.3.9) to the function v N , and adding it to the previous line. This yields

b N (v N , p 0 N q 0 N ) = ↵ Z ⌦ (u u N ) • v N dx + ↵ Z ⌦ u N • v N dx a N ↵ (u N v N ) + Z ⌦ v N • grad (P 1 q 0 N )dx + Z ⌦ v N • grad ( P p b N )dx +((↵ f, v N )) N ↵ Z ⌦ f • v N dx.
(3.5.73)

According to the triangle inequality, we have

||p q N || ↵ ⇤ = ||P 1 q 0 N || ↵ ⇤ + ||p 0 N q 0 N || ↵ ⇤ + || P p b N || ↵ ⇤ (3.5.74)
So using the same arguments as in the estimation of terms issued from numerical integration together with the above triangle inequality yields

||p q N || ↵ ⇤  c(||u u N || ↵ + sup v N 2X N ↵ R ⌦ u N • v N dx a N ↵ (u N , v N ) ||v N || ↵ +||P 1 q 0 N || ↵ ⇤ + sup v N 2X N ((↵ f, v N )) N ↵ R ⌦ f N • v N dx ||v N || ↵ + sup v N 2X N b(v N , P ) b N (v N , q 0 N ) ||v N || ↵ + || P p b N || ↵ ⇤ ).
(3.5.75)

Then we give the second theorem about the error estimate of the pressure:

Theorem 3.
||u u N || ↵ + ||p p N || ↵ ⇤  c(N m (||u|| H m (⌦) d + ||p|| H m+1 (⌦) ) +N r ||f|| H r (⌦) d + N s ||k|| H s ( 1 ) + N t ||p 0 || H t ( 2 ) ).
(3.5.76)

Numerical results

In this section, we will talk about the numerical computation for the schemes proposed in the above sections. Because ↵ is piecewise constant in our work, for simplicity, we suppose here a constant in the whole domain, and we set ↵ = 1.

For the details of the computation, including the linear system, Uzawa algorithm and the numerical implementation, we refer to ( [START_REF] Bernardi | Discrétisations variationnelles de problemes aux limites elliptiques[END_REF], V.4) and ([32], Chapter 2). Here we only give some numerical results to support our estimate convergence.For the simplicity of the computation, we set here 1 = @⌦. The : The relative error for the first case Where (u N , p N ) is the velocity and pressure respectively computed on the Legendre-Gauss-Labatto points. We can know that the error with order 10 15 . The following figure represents the logarithm of error, in norm L 2 (⌦) 2 and H 1 (⌦). It shows that the error decays fast when N increase. In addition, the error decays expontionaly when N increase. These results confirm the numerical analysis obtained in the above section. The Curve of the convergence in L 2 (+ ) and H 1 (+ ) The next table give the numerical results for the solution given above.

The first discretization

Norm L 2 (velocity) Norm H 1 (pressure)
||u u N || ↵,L 2 (⌦) ||u|| ↵,L 2 (⌦)
e-2 3e-6 4e-9 4.3e-12 2.8e-14

||p p N || ↵ ⇤ ||p|| ↵ ⇤
2.73e-4 5.49e-7 6e-10 3e-13 1.4e-14 Table : The relative error for the first case We can know that the error with order 10 14 . The following figure represents the logarithm of error, in norm L 2 (⌦) 2 and H 1 (⌦). It shows that the error decays fast when N increase. In addition, the error decays expontionaly when N increase. We The curve of convergence Norm L 2 (velocity) Norm H 1 (pressure) are also interested in the case where the solution of the problem is not so regular, so the third example as follows:

u(x, y) = 0 @ 5 2 y(x 2 + y 2 ) 1 4 5 2 x(x 2 + y 2 ) 1 4
1 A , p(x, y) = sin(⇡(x + y)).

(3.6.4)

The boundary conditions are:

k(x, y) = 8 < : 5 2 y(1 + y 2 ) 1 4
), if 1  y  1 and x = 1 and x = 1 The results are given in the following table:

N 5 9 13 17 19 ||u u N || ↵,L 2 (⌦) ||u|| ↵,L 2 (⌦)
e-3 7e-6 4e-9 6.3e-14 2.8e-15

||p p N || ↵ ⇤ ||p|| ↵ ⇤
2.4e-3 5.7e-5 e-8 3e-12 8.4e-15 Table : The relative error for the first case The next figure represents the logarithm of error, in norm L 2 (⌦) 2 and H 1 (⌦). It shows that the error decays fast when N increase. In addition, the error decays expontionaly when N increase.

In the end, we present the following problems, the knowns datas are: The curve of convergence Norm L 2 (velocity) Norm H 1 (pressure)

f(x, y) = 0 @ y 2 (1 x 2 ) 1 A , in ⌦, k(x, y) = 0 on @⌦ (3.

Introduction

Let ⌦ be a bounded, connected, open set in R d , d = 2 or 3, with a Lipschitz continuous boundary @⌦, and let n denote the unit outward normal vector to ⌦ on @⌦, 1 and 2 be a partition of @⌦ without overlop. We will talk about the Darcy's equations as follows

8 > > > > > > < > > > > > > : ↵u + grad p = ↵f in ⌦, div u = 0 in ⌦, u • n = k o n 1 , p = p 0 on 2 (4.1.1)
The data are now the function f and the boundary conditions k and p 0 . The unknowns are the velocity u and the pressure (or hydraulic head, according to the model) p.

We assume that the function ↵ is piecewise constant: There exists a partition of the domain ⌦ without overlap into connected subdomains ⌦ l , 1  l  L, such that ↵ is constant equal to ↵ l on ⌦ l . We introduce the notation

↵ min = min 1lL ↵ l and ↵ max = max 1lL ↵ l ,
and, without restriction, we assume that ↵ min is positive.

The mortar element method, due to Bernardi, Maday and Petera [START_REF] Bernardi | A new nonconforming approach to domain decomposition: the mortar element method[END_REF], is a domain decomposition technique which allows for working on general partitions of the domain, without conformity restrications. It is widely used to discrete second-order elliptic equations when set in standard Sobolev spaces, we refer to [START_REF] Bernardi | Mortar spectral element methods for elliptic equations with discontinuous coefficients[END_REF] and [START_REF] Azaïez | The mortar spectral element method in domains of operators. part I: The divergence operator and Darcy's equations[END_REF]. In [START_REF] Azaïez | The mortar spectral element method in domains of operators. part I: The divergence operator and Darcy's equations[END_REF], Azaïez, Belgacem and Bernardi extend this method to problems formulated in spaces of square-integrable functions with square-integrable divergence, where Darcy's equations with homogeneous Dirichlet boundary is discussed.

We firstly consider the key situation where the function ↵ is piecewise constant.

We refer to ( [START_REF] Barbara | Discretization Techniques Based on Domain Decomposition[END_REF], Subsec. 1.5) for a first application of this method to discontinuous coefficient in the finite element framwork. For spectral element framwork, we refer to [START_REF] Bernardi | Mortar spectral element methods for elliptic equations with discontinuous coefficients[END_REF] for the second-order elliptic equations and [START_REF] Daadaa | Discrétisation spectrale et par éléments spectraux des équations de Darcy[END_REF] for Darcy's equations formulated in standard Sobolev spaces. For the boundary conditions, we introduce a subspace of H(div , ⌦) with trace u • n = 0 on 1 . Thanks to the recent paper [START_REF] Bernard | Density results in Sobolev spaces whose elements vanish on a part of the boundary[END_REF] by Bernard, we can derive its density results and then the well-posedness of the problem.

The approximation properties of divergence-free functions in the mortar discrete space are needed for the numerical analysis of the discrete problem. According to this, wo can prove a spectral type error estimate: the order of convergence only depends on the regularity of the solution, more precisely on its local regularity in each subdomain.

Finally, an appropriate treatment of the matching conditions on the interface is the main point of the implementation of the mortar technique, we refer to [START_REF] Azaïez | The mortar spectral element method in domains of operators. part I: The divergence operator and Darcy's equations[END_REF]. We also refer to [START_REF] Faker | The mortar finite element method with Lagrange multipliers[END_REF] to see another way of treating these conditions. Two dimensional numerical experiments are presented. They are in good coherency with the numerical analysis.

Description of the mortar spectral element discretization

Throughout this section, we work with a piecewise constant function ↵. We now assume that ⌦ admits a disjoint decomposition into a finite number of (open) rectangles in dimension d = 2, rectangular parallelepipeds in dimension d = 3, denoted by ⌦ k , with edges all parallel to the coordinate axes,

⌦ = K [ k=1 ⌦k and ⌦ k \ ⌦ k 0 = ;, 1  k 6 = k 0  K. (4.2.1)
We make the further assumption that the intersection of each @⌦ k with @⌦, if not empty, is a corner, a whole edge or a whole face of ⌦ k . For 1  k  K, we denote by k,l , 1  l  L(k), the (open) edges in dimension d = 2, faces in dimension d = 3, of ⌦ k which are not contained in @⌦. We denote by n k the unit outward normal vector to ⌦ k on @⌦ k . The decomposition is said to be conforming when the intersection of two different ⌦k , if not empty, is a corner or a whole edge or a whole face of both of them. For simplicity, we denote by

↵ k the constant value of ↵ on each ⌦ k , 1  k  K.
The skeleton S of the decomposition, equal to K S k=1 @⌦ k \@⌦, admits a decomposition without overlapping into mortars

S = M [ m=1 ¯ m and m \ m 0 = ;, 1  m 6 = m 0  M. (4.2.2)
where each m = k(m),l(m) is a whole edge in dimension d = 2, face in dimension d = 3, of a subdomain ⌦ k , denoted by ⌦ k(m) . Note that the choice of this decomposition is not unique, however it is decided a priori for all the discretizations we work with.

In order to describe the discrete problem, we introduce the discretization parameter , which is here a K-tuple of positive integers

N k , 1  k  K, with each N k 2.
Indeed, the local discrete space on each ⌦ k is the space P N k (⌦ k ) of restrictions to ⌦ k of polynomials with degree  N k with respect to each variable. In all that follows, c stands for a generic constant which may vary from line to line but is always independent of .

We first introduce the discrete spaces. From the trace theory and (2.3.5) we know that the space X(⌦) coincides with the space of functions v such that their restrictions

to each ⌦ k , 1  k  K, belong to H(div , ⌦ k )
and their normal traces are continuous through the skeleton S. For 1  m  M , we define the mortar function ' on each

m by '| m = v | ⌦ k(m) • n k(m) .
Then we define the corresponding discrete space of velocities D (⌦) of functions v such that

1. their restrictions v | ⌦ k to each ⌦ k , 1  k  K, belong to P N k (⌦ k ) d ,
2. such the following matching condition holds on each edge k,l , 1  k  K, 1  l  L(k), which is not a mortar:

8 2 P N k 2 ( k,l ), Z k,l (v | ⌦ • n k + ')(t) (t)dt = 0. (4.2.3)
We also define the following space

D 0 (⌦) = {v 2 D (⌦)| v • n = 0 on 1 } (4.2.4)
According to ([8], Lemma 4.1) and ( [START_REF] Bernardi | A new nonconforming approach to domain decomposition: the mortar element method[END_REF], Thm 24.1), the space M of the discrete pressures is introduced by:

M (⌦) = {q 2 L 2 (⌦); q | ⌦ k 2 P N k 2 (⌦ k ), 1  k  K}. (4.2.5)
Assuming that the functions f, k and p 0 have continuous restrictions to all ⌦k , 1  k  K and ¯ 1 , ¯ 2 respectively. Then the discrete problem built from the variational formulation (2.3.7) by the Galerkin method with numerical integration(GNI) reads :

Find (u , p ) in D (⌦) ⇥ M (⌦) such that u • n = I 1 k on 1 , (4.2.6)
and

8v 2 D 0 (⌦), a ↵ (u , v ) + b (v , p ) = L (v ), 8q 2 M (⌦), b (u , q ) = 0. (4.2.7)
where the bilinear forms a ↵ (•, •) and b are defined by

a ↵ (u , v ) = K P k=1 ↵ k ((u , v )) k , b (v , q ) = K P k=1 ((div v , q )) k , L (v ) = K P k=1 ↵ k ((f , v )) k ((p 0 , v • n)) 2 . (4.2.8)
Thanks to the exactness property of the quadrature formula, we note that

8v 2 D 0 (⌦), 8q 2 M (⌦), b (v , q ) = Z ⌦ q (div v )dx. (4.2.9)
Since D (⌦) is not contained in H(div , ⌦) in the general case, we need the "bro-Lemma 4.4 There exists an integer N D only depending on the decomposition of ⌦ such that, if all the N k N D , the following inf-sup condition holds

8q 2 M (⌦), sup v 2D 0 (⌦) R ⌦ q (div v )dx ||v || ↵, D ||q || ↵ ⇤ , , (4.2.18)
for a positive constant D depending only on the decomposition of ⌦ but not .

Proof: Let q be any function in M (⌦), according to the Boland and Nicolaides technique [START_REF] Boland | Stability of finite elements under divergence constraints[END_REF], the idea consists in writing q as q = q + q , with q

| ⌦ k = 1 meas(⌦ k ) Z ⌦ k q (x)dx, 1  k  K.
The decomposition is orthogonal in the sense of

||q || 2 L 2 (⌦) = ||q || 2 L 2 (⌦) + ||q || 2 L 2 (⌦) . (4.2.19)
Since q | ⌦ k has a null integral on ⌦ k , it follows from [START_REF] Azaïez | Spectral methods applied to porous media[END_REF] that there exists a function

v k in P N k T H 0 (div , ⌦ k ) such that div v k = q | ⌦ k and ||v k || H(div ,⌦ k )  c||q | ⌦ k || L 2 (⌦) . (4.2.20)
So, the function ṽ defined by v

| ⌦ k = v k , 1  k  K, belongs to D 0 (⌦) (and even to H 0 (div , ⌦)).
Concerning the function q , (note that its integral on ⌦ is equal to the integral of q , hence to zero) , we refer to ( [START_REF] Faker | Inf-sup conditions for the mortar spectral element discretization of the Stokes problem[END_REF], Proposition 3.1) that there exists at least a function

v in D D , with v | ⌦ k in P N D (⌦ k ) d , such that div v = q and ||v || H(div ,⌦)  c||q || L 2 (⌦) . ( 4

.2.21)

This function v belongs to D 0 (⌦) when all the N k N D .

We finally take: v = ṽ + µv , for a positive parameter µ. Indeed, div ṽ is orthogonal to q (this comes from the Stokes formula applied on each ⌦ k and grad q = 0), so that, from the previous two equations, we have

Z ⌦ q (div v )dx ||q || 2 L 2 (⌦) + µ||q || 2 L 2 (⌦) µ||div v || L 2 (⌦) ||q || L 2 ⌦ .
Then it is readily checked that

Z ⌦ q (div v )dx ||q || 2 L 2 (⌦) + µ||q || 2 L 2 (⌦) µc||q || L 2 (⌦) ||q || L 2 ⌦ (4.2.22) 1 2 ||q || 2 L 2 (⌦) + µ(1 µc 2 2 )||q || 2 L 2 (⌦) . (4.2.

23)

Choosing µ = 1 c2 and using (4.2.19) gives

Z ⌦ q (div v ) c||q || 2 L 2 (⌦) .
On the other hand ,we have

||v || ↵,  c(||q || L 2 (⌦) + µ||q || L 2 ⌦ )  c||q || ↵ ⇤ , .
This ends the proof.

It follows from Lemma 4.2 and Lemma 4.4 that problem (4.2.13) has a unique solution (u 0 , p ) in D 0 (⌦) ⇥ M (⌦) and that this solution satisfies

||u 0 || ↵, + ||p || ↵ ⇤ ,  C(||u b || ↵, + ||f|| D 0 (⌦) 0 + ||p 0 || H 1 2 00 ( 2 ) ) (4.2.24)
We now have the main result of this section.

Approximation of divergence-free functions.

We now intend to estimate the distance of a divergence-free function to the space • equal to the maximum of the ratios ↵ l /↵ l 0 in the case of a conforming decomposition,

V (⌦) introduced in (4.
• equal to the maximum of the ratios (↵ l N l )/(↵ l 0 N l 0 ) for all adjacent subdomains ⌦ l and ⌦ l 0 , 1  l, l 0  L, where ⌦ l and ⌦ l 0 share a part of an edge in dimension d = 2, or a part of a face in dimension d = 3.

We also define the quantity µ as the largest ratios of ↵ l /↵ l 0 .

We note that the constant µ only depends on the decomposition and on the choice of the mortars but not on the discretization parameter. However µ depends on for a nonconforming decomposition.

We introduce here new norms as follows:

||v|| ↵,L 2 (⌦) = ( K X k=1 ↵ k ||v |⌦ k || 2 L 2 (⌦ k ) ) 1 2 ||v|| ↵,H 1 (⌦) = ( K X k=1 ↵ k ||v |⌦ k || 2 H 1 (⌦ k ) ) 1 2
there exists a constant c independent of such that

inf ⇠ ⇠ ⇠ 2C (⌦) ||u curl ⇠ ⇠ ⇠ || ↵,L 2 (⌦) 3  c(1 + µ 1 2 )( K X k=1 ↵ k N 2s k k ||u|| 2 H s k (⌦ k ) 3 ) 1 2 (4.2.

32)

Proof: It follows from ( [START_REF] Faker | Spectral element discretization of the Maxwell equations[END_REF], Them 4.9) that for

1  k  K and s k 3 2 , there exists a polynomial ⇠ ⇠ ⇠ k in P N k (⌦) 3 such that ⇠ ⇠ ⇠ k ⇥ n coincide with ⇠ ⇠ ⇠ k ⇥ n and that ↵ k ||curl ⇠ ⇠ ⇠ curl ⇠ ⇠ ⇠ k || L 2 (⌦ k )  c↵ k N s k k ||u|| H s k (⌦ k ) 3 . (4.2.33) Morever, for 1  l  L(k), this function ⇠ ⇠ ⇠ k satisfies 8g 2 P N k 2 ( k,l ) 3 , Z k,l (⇠ ⇠ ⇠ ⇠ ⇠ ⇠ k ) ⇥ n • gd⌧ ⌧ ⌧ = 0. (4.2.34)
In the case of a conforming decomposition and thanks to the assumption on the N k , equation (4.2.34) implies that the function ⇠ ⇠ ⇠ defined by

⇠ ⇠ ⇠ |⌦ k = ⇠ ⇠ ⇠ k , 1  k  K,
belongs to C , whence the result.

Proposition 4.3 Assume that the function u in V such that each u |⌦ k , 1  k  K, belongs to H s k (⌦ k ) d , s k 3 2 .
In the case of dimension d = 3, if the decomposition is conforming and if morever for each mortar m , 1  m  M, which is a face of both

⌦ k(m) and ⌦ k , N k(m) N k , there exists a constant c independent of such that inf v 2V (⌦) ||u v || ↵,L 2 (⌦) 3  c(1 + µ 1 2 )( K X k=1 ↵ k N 2s k k ||u|| 2 H s k (⌦ k ) 3 ) 1 2 (4.2.

35)

Proof: It follows from [START_REF] Amrouche | Vector potentials in three-dimensional non-smooth domains[END_REF] that there exists a vector potential ⇠ such that u = curl ⇠. Estimate (4.2.32) yields that the function v = curl ⇠ satisfies the desired property (4.2.35). Moreover, it can be checked ( [START_REF] Faker | Spectral element discretization of the Maxwell equations[END_REF], Chapter 4) that, on each face of ⌦ k , curl ⇠ • n is equal to the orthogonal projection (in L 2 ( )) of u • n onto P N k 1 ( ). So the jump of curl ⇠ • n through is zero (or its trace if is contained in @⌦), and the function v belongs to V (⌦).

Error estimate

We prove an error estimates, first for the velocity, second for the pressure. Let u be the solution of equation ( 4

K X k=1 ↵ k Z ⌦ k u 0 • ! ! ! dx + b(! ! ! , p) = K X k=1 ↵ k Z ⌦ k f • ! ! ! dx K X k=1 ↵ k Z ⌦ l u b • ! ! ! dx Z 2 p • ! ! ! • nds Z S p • [! ! ! • n]d⌧ ⌧ ⌧
where the notation [•] means the jump across S.

Using the definition of V (⌦) thus implies, for any q in M (⌦),

K X k=1 ↵ k Z ⌦ k u 0 • ! ! ! dx + b(! ! ! , p q ) = K X k=1 ↵ k Z ⌦ k f • ! ! ! dx K X k=1 ↵ k Z ⌦ l u b • ! ! ! dx Z 2 p • ! ! ! • nds Z S p • [! ! ! • n]d⌧ ⌧ ⌧ (4.2.36)
Due to (??), we have for any v in V

||u 0 v || 2 ↵,L 2 (⌦)  K X k=1 ↵ k ((u 0 v , u 0 v )) k .
In (4.2.36), letting ! ! ! = u 0 v and subtracting the first line of (4.2.7) leads to

||u 0 v || 2 ↵,L 2 (⌦)  K X k=1 ↵ k Z ⌦ k (u 0 v ) • (u 0 v )dx + K X k=1 ↵ k Z ⌦ k v • (u 0 v )dx K X k=1 ↵ k ((v , u 0 v )) k + b(u 0 v , p q ) + Z S [(u 0 v ) • n] • pd⌧ ⌧ ⌧ + K X k=1 ↵ k ((f, u 0 v )) k K X k=1 ↵ k Z ⌦ k f • (u 0 v )dx + K X k=1 ↵ k Z ⌦ k u b • (u 0 v )dx K X k=1 ↵ k ((u b , u 0 v )) k Z 2 p 0 • [(u 0 v ) • n]ds ((p 0 , (u 0 v ) • n)) 2 .
Using a triangle inequality yields

||u u || ↵,L 2 (⌦)  ||u 0 v || ↵,L 2 (⌦) + ||u 0 v || ↵,L 2 (⌦) + ||u b u b || ↵,L 2 (⌦)  C{ inf v 2V ||u 0 v || ↵,L 2 (⌦) + sup ! ! ! 2D { K P k=1 ↵ k R ⌦ k v • ! ! ! dx K P k=1 ↵ k ((v , ! ! ! )) k ||! ! ! || ↵,L 2 (⌦) + K P k=1 ↵ k ((f, ! ! ! )) k K P k=1 ↵ k R ⌦ k f • ! ! ! dx ||! ! ! || ↵,L 2 (⌦) + K P k=1 ↵ k R ⌦ k u b • ! ! ! dx K P k=1 ↵ k ((u b , ! ! ! )) k ||! ! ! || ↵,L 2 (⌦) + R 2 p 0 • (! ! ! • n)ds ((p 0 , (! ! ! • n)) 2 ||! ! ! || ↵,L 2 (⌦) + b(! ! ! , p q ) ||! ! ! || ↵,L 2 (⌦) + R S [! ! ! • n] (p q )d⌧ ⌧ ⌧ ||! ! ! || ↵,L 2 (⌦) + ||u b u b || ↵,L 2 (⌦) }. (4.2.37)
Next, let ⇧ ⇧ ⇧ N k 1 be defined as the orthogonal projection operator from L 2 (⌦ k ) onto

P N k 1 (⌦ k )
. By adding and subtracting the function ⇧ ⇧ ⇧ N k 1 u 0 , we deduce from the exactness of the quadrature formula that

K X k=1 ↵ k Z ⌦ k v • ! ! ! dx K X k=1 ↵ k ((v , ! ! ! )) k = K X k=1 ↵ k Z ⌦ k (v ⇧ ⇧ ⇧ N k 1 u 0 ) • ! ! ! dx K X k=1 ↵ k ((v ⇧ ⇧ ⇧ N k 1 u 0 ), ! ! ! )) k (4.2.38) So we have sup ! ! ! 2D K P k=1 ↵ k R ⌦ k v • ! ! ! dx K P k=1 ↵ k ((v , ! ! ! )) k ||! ! ! || ↵,L 2 (⌦)  (3 d + 1)( K X k=1 ↵ k ||v ⇧ ⇧ ⇧ N k 1 u 0 || 2 L 2 (⌦ k ) ) 1 2  (3 d + 1)||u 0 v || ↵,L 2 (⌦) + (3 d + 1)( K X k=1 ↵ k ||u 0 ⇧ ⇧ ⇧ N k 1 u 0 || 2 L 2 (⌦ k ) ) 1 2 .
Suppose the solution u 0 |⌦ k is in H s k (⌦ k ) d , s k 0, then according to the properity of the orthogonal projection operator,

sup ! ! ! 2D K P k=1 ↵ k R ⌦ k v • ! ! ! dx K P k=1 ↵ k ((v , ! ! ! )) k ||! ! ! || ↵,L 2 (⌦)  4||u 0 v || ↵,L 2 (⌦) + c( K X k=1 ↵ k N 2s k k ||u 0 |⌦ k || 2 H s k (⌦ k ) d ) 1 2 .
(4.2.39)

Similarly, we also have

K X k=1 ↵ k Z ⌦ k f • ! ! ! dx K X k=1 ↵ k ((f, ! ! ! )) k = K X k=1 ↵ k Z ⌦ k (f ⇧ ⇧ ⇧ N k 1 f) • ! ! ! dx K X k=1 ↵ k ((f ⇧ ⇧ ⇧ N k 1 f), ! ! ! )) k  p ↵ max (10( K X k=1 ||f ⇧ ⇧ ⇧ N k 1 f|| 2 L 2 (⌦ k ) d ) 1 2 + 9||f I f|| L 2 (⌦ k ) d )||! ! ! || L 2 (⌦ k ) d .
sup ! ! ! 2D K P k=1 ↵ k ((f, ! ! ! )) k K P k=1 ↵ k R ⌦ k f • ! ! ! dx ||! ! ! || ↵,L 2 (⌦)  c p ↵ max ( K X k=1 N 2 k k ||f |⌦ k || 2 H k (⌦ k ) d ) 1 2 . (4.2.41)
Similarly suppose the function p 0 is in

H ✓ k ( 2 \ @⌦ k ), ✓ k 0, then we have sup ! ! ! 2D R 2 p 0 • (! ! ! • n)ds ((p 0 , (! ! ! • n)) 2 ||! ! ! || ↵,L 2 (⌦)  c r 1 ↵ min ( K 0 X k=1 N 2✓ k k ||p 0 || 2 H ✓ k ( 2 \@⌦ k ) ) 1 2 (4.2.42)
We refer to ( [START_REF] Bernardi | Spectral methods[END_REF], Thm.7.1) and ( [START_REF] Bernardi | Spectral methods[END_REF], Thm.14.2) for the approximation properties of the operators ⇧ ⇧ ⇧ N k 1 and I , respectively.

According to Lemma 4.1, we also obtain that

sup ! ! ! 2D K P k=1 ↵ k R ⌦ k u b • ! ! ! dx K P k=1 ↵ k ((u b , ! ! ! )) k ||! ! ! || ↵,L 2 (⌦)  c p ↵ max ( K X k=1 N 2s k k ||u b |⌦ k || 2 H s k (⌦ k ) d ) 1 2 .
(4.2.43)

According to ([6], Lemma 6.1 and (6.9)) we know that

Z S [! ! ! • n](p q )d⌧ ⌧ ⌧  c r 1 ↵ min (1 + µ)( K X k=1 L(k) X l=1 N 2 k ||p q || 2 L 2 ( k,l ) ) 1 2 ||! ! ! || ↵,L 2 (⌦) . (4.2.44) So we have sup ! ! ! 2D R S [! ! ! • n](p q )d⌧ ⌧ ⌧ ||! ! ! || ↵,L 2 (⌦)  c r 1 ↵ min (1 + µ)( K X k=1 L(k) X l=1 N 2 k ||p q || 2 L 2 ( k,l ) ) 1 2 . (4.2.45)
the estimation of norm ||p q || 2 L 2 ( k,l ) will be showed later.

Because of the ellipticity property of a

↵ (•, •) in V , we have b(! ! ! , p q ) = K X k=1 Z ⌦ k div (! ! ! )(p q )dx  ||div (! ! ! )|| ↵,L 2 (⌦) • K X k=1 1 p ↵ min ||p q || L 2 (⌦ k ) (4.2.46) So we can get sup ! ! ! 2D b(! ! ! , p q ) ||! ! ! || ↵,L 2 (⌦)  c K X k=1 1 p ↵ min ||p q || L 2 (⌦ k ) . (4.2.47)
Next, we choose q by: q

|⌦ k = ⇧ ⇧ ⇧ 1 N k 2 p, where ⇧ ⇧ ⇧ 1 N k 2 stands for the orthogonal projection operator from H 1 (⌦ k ) onto P N k 2 (⌦ k ) defined by 8q 2 P N k 2 (⌦ k ), Z ⌦ k grad(p ⇧ ⇧ ⇧ 1 N k 2 p) • gradqdx = 0, Z ⌦ k (p ⇧ ⇧ ⇧ 1 N k 2 p)dx = 0
In fact,the following approximation properties of this operator are well-known: if the

function p |⌦ k belongs to H s k +1 (⌦ k ), s k 0, ||p ⇧ ⇧ ⇧ 1 N k 2 p|| H 1 (⌦ k ) + N k ||p ⇧ ⇧ ⇧ 1 N k 2 p|| L 2 (⌦ k )  cN s k k ||p|| H s k +1 (⌦ k ) . (4.2.48) While estimating the norm ||p ⇧ ⇧ ⇧ 1 N k 2 p|| L 2 ( k,l
) results from a duality argument and we refer to ([6], Lemma 6.2) for the following lemma.

Lemma 4.6 For 1  k  K, there exists a constant c independent of N such that, if the function p| ⌦ k belongs to H s k +1 (⌦ k ), s k 0, ||p ⇧ ⇧ ⇧ 1 N k 2 p|| L 2 ( k,l )  cN s k 1 2 k ||p|| H s k +1 . (4.2.49)
We are in a position to write the first error estimate, due to the above analysis and using the approximation properties stated in Propositions 4.2 and Propositions 4.3.

Theorem 4.1 Assume the data f such that each f| ⌦ k , 1  k  K, belongs to H k (⌦ k ) d , k > d 2 , the solution (u, p) of problem (4.1.1) such that each (u| ⌦ k , p| ⌦ k ), 1  k  K, belongs to H s k (⌦ k ) d ⇥ H s k +1 (⌦ k ), s k d 3 2 .
Then, in the two cases

1. in dimension d = 2,
2. in dimension d = 3, if the decomposition is conforming and if morever for each

mortar m , 1  m  M, which is a face of both ⌦ k(m) and ⌦ k , N k(m) N k ,
the following error estimate holds betweem the velocity u and the velocity u of problem (4.2.6) (4.2.7):

||u u || ↵,L 2 (⌦)  c K X k=1 {(1 + µ + µ ) p ↵ max N s k k ||u |⌦ k || H s k (⌦ k ) +(1 + µ ) 1 p ↵ min N s k + 1 2 k ||p |⌦ k || H s k +1 (⌦ k ) + p ↵ max N k k ||f |⌦ k || H k (⌦ k ) } +c K 0 X k=1 1 p ↵ min N s k 1 2 k ||p 0 || H s k + 1 2 ( 2 \@⌦ k ) +c K 1 X k=1 (1 + µ + µ ) p ↵ max N s k k ||k|| H s k 1 2 ( 1 \@⌦ k ) . (4.2.50)
Remark 4.1 Estimate (4.2.50) is not fully optiaml, however, we know that for a general decomposition (of course inculding the nonconforming case), when

(1 + µ + µ ) max{ p ↵ max , 1 p ↵ min } is bounded (in fact
, this is most often the case in practical situations), the lack of optimality is only N

1 2
k . In fact, the estimate results of the two cases when the decomposition is conforming, seems to be improved, but we don't prove the results here.

A more explicit estimate can be deduced from the previously quoted regularity results in the two-dimension case of a polygon ⌦. We refer to ([6], Corollary 6.4) and [START_REF] Bernardi | Discrétisations variationnelles de problemes aux limites elliptiques[END_REF] for the next result.

Corollary 4.1 In the two-dimensional case of a polygon ⌦, we have the same assumptions with T heorem4.1 about the data f . Then the following error estimate holds between the velocity u and the velocity u of problem (4.2.6) (4.2.7):

||u u || ↵,L 2 (⌦)  c(1 + µ + µ ) max{ p ↵ max , 1 p ↵ min } K X k=1 N 1 2 k E k ||f |⌦ k || H k (⌦ k ) +c K 0 X k=1 1 p ↵ min N 1 2 k E k ||p 0 || H k + 1 2 ( 2 \@⌦ k ) +c(1 + µ + µ ) max{ p ↵ max , 1 p ↵ min } K 1 X k=1 E k ||k|| H k 1 2 ( 1 \@⌦ k )
where

E k , 1  k  K is equal to N k k if ⌦k contains no corner of ⌦, sup{N k k , N 4 3 k (log N k ) 1 2 }, if ⌦k contains a nonconvex corner of ⌦ and sup{N k k , N 4 k (log N k ) 3 2 }, if ⌦k contains a no nonconvex corner of ⌦.
Next we will estimate the error of the pressure which is easier now.

Theorem 4.2 If the asumptions of Theorem 5.1 are satisfied and in case (i) and (ii) of this theorem, the following error estimate holds between the pressure p of problem (4.1.1) and the pressure p of (4.2.7):

||p p || ↵ ⇤ ,  c K X k=1 {(1 + µ + µ ) p ↵ max N s k k ||u |⌦ k || H s k (⌦ k ) +(1 + µ + µ ) max{ p ↵ max , 1 p ↵ min }N s k k max{N 1 2 k , (log N k ) ⌫ }||p |⌦ k || H s k +1 (⌦ k ) + p ↵ max N k k ||f |⌦ k || H k (⌦ k ) } + c K 0 X k=1 1 p ↵ min N s k 1 2 k ||p 0 || H s k + 1 2 ( 2 \@⌦ k ) +c K 1 X k=1 (1 + µ + µ ) p ↵ max N s k k ||k|| H s k 1 2 ( 1 \@⌦ k ) .
where ⌫ is equal to zero if the decomposion is conforming, to 1 otherwise.

Proof: Due to the inf sup condition, for any q in M (⌦), we derive that

D ||p q || ↵ ⇤ ,  sup v 2D 0 (⌦) R ⌦ (p q )(div v )dx ||v || ↵, . (4.2.51)
In order to evaluate R ⌦ (p q )(div v )dx, we first use the discrete problem (4.2.7):

Z ⌦ (p q )(div v )dx = K X k=1 ↵ k ((f, v )) k ((p 0 , v • n)) 2 K X k=1 ↵ k ((u , v )) k + Z ⌦ q (div v )dx. (4.2.52)
Next, we apply equation (4.1.1) to the function v , integrate by parts and add it to the previous line, which yields

Z ⌦ (p q )(div v )dx = K X k=1 ↵ k ((f, v )) k K X k=1 ↵ k Z ⌦ k f • v dx + K X k=1 ↵ k Z ⌦ k u • v dx K X k=1 ↵ k ((u, v )) k + Z 2 p 0 • (v • n) ds ((p 0 , v • n)) 2 + Z ⌦ (q p) div v dx + Z S [v • n] • pd⌧ ⌧ ⌧ .
Using a triangle inequality yields

||p p || ↵ ⇤ ,  c(||p q || ↵ ⇤ , + ||u u || ↵,L 2 (⌦) + sup ! ! ! 2D 0 { K P k=1 ↵ k R ⌦ k u v dx K P k=1 ↵ k ((u , v )) k ||v || ↵,L 2 (⌦) + K P k=1 ↵ k ((f, v )) k K P k=1 ↵ k R ⌦ k f • v dx ||v || ↵,L 2 (⌦) + R 2 p 0 • (v • n)ds ((p 0 , (v • n)) 2 ||v || ↵,L 2 (⌦) + R S [v • n]pdx ||v || ↵, }). (4.2.53)
All the right-hand side terms have been estimated previously, except the last one which corresponds to the consistency error. We refer to ( [START_REF] Bernardi | Mortar spectral element methods for elliptic equations with discontinuous coefficients[END_REF], Lemma 3.3) for the following result:

sup ! ! ! 2D 0 R S [v • n]pdx ||v || ↵,  c(1 + µ)( K X k=1 ↵ k N 2s k k (log N k ) ⌫ ||p|| 2 s k +1 ) 1 2 . (4.2.54)
where ⌫ is equal to zero if the decomposion is conforming, to 1 otherwise.

Combining all the results above, yields the desired estimate.

As in Corollary 4.1, we also have the following result:

Corollary 4.2 In the two-dimensional case of a polygon ⌦, if the same assumptions as in T heorem4.1 hold about the data f . Then the following error estimate holds between the velocity p and the velocity p of problem (4.2.7):

||p p || ↵ ⇤ ,  c(1 + µ + µ ) max{ p ↵ max , 1 p ↵ min } K X k=1 max{N 1 2 k , (log N k ) ⌫ }E k ||f |⌦ k || H k (⌦ k ) +c K 0 X k=1 1 p ↵ min N 1 2 k E k ||p 0 || H k + 1 2 ( 2 \@⌦ k ) +c(1 + µ + µ ) max{ p ↵ max , 1 p ↵ min } K 1 X k=1 E k ||k|| H k 1 2 ( 1 \@⌦ k ) (4.2.55)
where ⌫ is equal to zero if the decomposion is conforming, to 1 otherwise; and E k , 1  k  K are given by:

E k = 8 > > > < > > > : N k k , if ⌦k contains no corner of ⌦, sup{N k k , N 4 k (log N k ) 3 2 }, if ⌦k contains a no nonconvex corner of ⌦, sup{N k k , N 4 3 k (log N k ) 1 2 }, if ⌦k contains a nonconvex corner of ⌦. (4.2.56)

Numerical implementation

Firstly, we briefly explain how to implement the discrete problem in section before.

The full set of unknowns of the discrete system is given by

• the vector U of the values if u at all nodes (x k i , y k i ), 0  i, j  N k , 1  k  K, in dimension d = 2, (x k i , y k i , z k p ), 0  i, j, p  N k , 1  k  K, in dimension d = 3,
• the vector P of the values if p at all nodes

(x k i , y k i ), 0  i, j  N k , 1  k  K, in dimension d = 2, (x k i , y k i , z k p ), 0  i, j, p  N k , 1  k  K, in dimension d = 3.
Let (U, P ) denote the vector made of these values. Then the mortar condition can be expressed in the following way: there exists a rectangular matrix Q such that the boundary and matching conditions can be enforced, which leads to a new vector QU .

Then problem (

) can now equivalent to the following linear system

0 @ Q T AQ Q T B B T Q 0 1 A 0 @ U P 1 A = 0 @ Q T F Q 0 1 A , (4.3.1)
where Q T stands for the transposed matrix of Q. The matrix A is fully diagonal, its diagonal terms are the ⇢ x,k i , ⇢ y,k j or the ⇢ x,k i , ⇢ y,k j , ⇢ z,k p according to the dimension. The matrix B is only block-diagonal, with K blocks B k on the diagonal, one for each ⌦ k .

For the first experiment, we concern the 2D L shaped domain

⌦ = [ 1, 3] 2 \ [1, 3] 2 ,
partitioned into three subdomains illustrated in Fig. 4-1 (upper panel). 1 will be chosen to be

[ 1, 1] ⇥ {3} [ { 1} ⇥ [ 1, 1], then 2 = @⌦ \ 1 .
We divide the three domains as follows:

⌦ 1 = [ 1, 1] ⇥ [1, 3], ⌦ 2 = [ 1, 1] ⇥ [ 1, 1], ⌦ 3 = [1, 3] ⇥ [ 1, 1],
which associated with N 1 = 22, N 2 = 17, N 3 = 17. We set here ↵ 1 = 1, ↵ 2 = 10, ↵ 3 = 100. We use our spectral method to compute an approximation of the analytical solution (u, p) given by u(x, y) = 0 @ sin(⇡x) cos(⇡y) cos(⇡x) sin(⇡y)

1 A , p(x, y) = sin(⇡(x + y)). (4.3.2)
The lower panel of Fig. 4-1 depicts, in a semi-logarithmic scale, the curves of the errors ||u u || ↵,L 2 (⌦) and ||p p || ↵ ⇤ ,L 2 (⌦) as a function of N .As can be seen from estimates in section 5, the convergence rate here is nearly exponential despite the non-conformity of the discretization.

We then consider the square ⌦ = [ 1, 3] 2 , with a nonconforming decomposition into

two squares ⌦ 1 = [ 1, 1] 2 , and ⌦ 3 = [1, 3]⇥[ 1, 1] and a rectangle ⌦ 2 = [ 1, 3]⇥[1, 3].
For an integer N 8, we take all the N k equal to N . We work here with the solutions (u, p) given by u( Mortar Spectral Discretization of Darcy's Equations(II)

Introduction

In this chapter, we mainly talk about the analysis of the mortar spectral element discretization of the problem introduced by Darcy [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF] as follows

8 > > > > > > < > > > > > > : ↵u + grad p = ↵f in ⌦, div u = 0 in ⌦, u • n = k o n 1 , p = p 0 on 2 (5.1.1)
where ⌦ be a bounded, connected, open set in R d , d = 2 or 3, with a Lipschitz continuous boundary @⌦, and let n denote the unit outward normal vector to ⌦ on @⌦, 1 and 2 be a partition of @⌦ without overlop. The data are now the function f and the boundary conditions k and p 0 . The unknowns are the velocity u and the pressure (or hydraulic head, according to the model) p. we are interested in the case where this function is not globally continuous but only piecewise smooth and also such that the ratio of its maximal value to its minimal value is large.

This problem is an expansion of [START_REF] Daadaa | Discrétisation spectrale et par éléments spectraux des équations de Darcy[END_REF], where the problem with special boundary condition has been discussed. Here the idea of this paper is different: our problem has the general boundary conditions, including u and p. So new spaces with special traces have to be introduced.

Analysis of the mortar SE discretization

Throughout this section, we work with a piecewise constant function ↵. We now assume that ⌦ admits a disjoint decomposition into a finite number of (open) rectangles in dimension d = 2, rectangular parallelepipeds in dimension d = 3, denoted by ⌦ k , with edges all parallel to the coordinate axes,

⌦ = K [ k=1 ⌦k and ⌦ k \ ⌦ k 0 = ;, 1  k 6 = k 0  K. (5.2.1)
We make the further assumption that the intersection of each @⌦ k with @⌦, if not empty, is a corner, a whole edge or a whole face of ⌦ k . For 1  k  K, we denote by k,l , 1  l  L(k), the (open) edges in dimension d = 2, faces in dimension d = 3, of ⌦ k which are not contained in @⌦ . We denote by n k the unit outward normal vector to ⌦ k on @⌦ k . The decomposition is said to be conforming means that the intersection of two different ⌦k , if not empty, is a corner or a whole edge of both of them.

The skeleton S of the decomposition, equal to We also introduce another discrete space M 1 associated with M :

M 1 = {q 2 M | q = 0 on 2 }.
(5.2.5)

Assuming that the functions f, k and p 0 has continuous restrictions to all ⌦k , 1  k  K and ¯ 1 , ¯ 2 respectively. Then the discrete problem built from the variational formulation(2.3.9) reads:

Find (u , p ) in X (⌦) ⇥ M (⌦) such that p = I 2 p 0 on 2 , (5.2.6)
and

8v 2 X (⌦), a ↵ (u , v ) + b (v , p ) = L (v ), 8q 2 M 1 (⌦), b (u , q ) = ((k, q )) 1 .
(5.2.7)

where the bilinear forms a ↵ (•, •) and b are defined by

a ↵ (u , v ) = K P k=1 ↵ k ((u , v )) k , b (v , q ) = K P k=1 ((v , grad q )) k , L (v ) = ((↵ f, v )) .
(5.2.8)

Since M is not contained in H 1 (⌦), we define the "broken" norm and semi-norm

||v || ↵ = ( K X k=1 ↵ k ||u || 2 L 2 (⌦ k ) d ) 1 2 , ||q || ↵ ⇤ = ( K X k=1 ↵ 1 k |q | 2 H 1 (⌦ k ) ) 1 2
Let M be the maximal number of points of which are contained in the same edge k,j . We first introduce the following space M 1 D of functions q definedon ⌦, such that:

1. which belongs to L 2 (⌦),

their restriction to each

⌦ k belongs to H 1 (⌦ k ),1  k  K, 3. vanish on 2 ,
4. the following matching condition holds on all subdomains ⌦ k , 1  k  K, and for all k,j of ⌦ k that are not contained in @⌦ and not a mortar,

8' 2 P M 2 ( k,j ), Z k,j (q |⌦ k )'d⌧ = 0, (5.2.9)
where P M 2( k,j ) is the space of polynomials with degree  M 2 on k,j .

Not that the definition of M D is similar to that of M 1 , however M 1 D is no longer a discrete space: its dimension is infinite and it only depends on the decomposition of the domain. Not also that M 1 D contains all the spaces M 1 , such that the N k in are larger than M . Lemma 5.1 There exists a constant C independent of such that

8q 2 M 1 D (⌦), K X k=1 ||q|| 2 L 2 (⌦ k )  C p ↵ max ||q|| 2 ↵ ⇤ .
(5.2.10)

and C is only dependent with the decomposition of ⌦.

Proof: Firstly, we consider a function q in M 1 D such that all the right hand side of (5.2.10) is zero. Thus, q is a constant on each ⌦ k and due to the choice of M , according to ( [START_REF] Bernardi | Spectral element method and mortar element element method in theory and numericals of differential equations[END_REF], Proposition 20), which yield that q is zero on ⌦. Next, we observe that

||q|| H 1 ( S ⌦) = c(||q|| 2 L 2 (⌦) + K X k=1 Z ⌦ k (grad q |⌦ k ) 2 dx) 1 2 .
(5.2.11) Also, since the imbedding of each So, combining the above inequality and triangle inequality, we derive that the error ||u u || ↵ is bounded, up to a multiplicative constant, by the sum of seven terms:

H 1 (⌦ k ) in L 2 (⌦ k ) is compact, the imbedding of M 1 D into L 2 (⌦) is compact. So,
• the approximation error in X

inf v 2V ||u 0 v || ↵ , (5.3.6)
• the error approximation in

M 1 inf q 2M 1 ||P 1 q || ↵ ⇤ , (5.3.7)
• three terms issued from numerical integration

sup ! ! ! 2X a ↵ (v , ! ! ! ) a ↵ (v , ! ! ! ) ||! ! ! || ↵ , (5.3.8)
sup We first estimate the terms issured from numerical integration, which is easy since they can be evaluated seperately on each subdomain ⌦ k . Let ⇧ N k 1 denote the orthogonal projection operator from L 2 (⌦ k ) onto

! ! ! 2X a ↵ (u b , ! ! ! ) a ↵ (u b , ! ! ! ) ||! ! ! || ↵ , (5.3.9)
((↵f, ! ! ! )) ↵ K P k=1 ↵ k R ⌦ k f • ! ! ! dx ||! ! ! || ↵ , (5.3
P N k 1 (⌦ k ) for each k, 1  k  K.
For any ! ! ! in X , since each product of ⇧ N k 1 u by ! ! ! belongs to P 2N k 1 (⌦ k ), it follows from the exactness propertity (??) that

a ↵ (v , ! ! ! ) a ↵ (v , ! ! ! ) = K X k=1 ↵ k ( Z ⌦ k (v ⇧ N k 1 u 0 ) • ! ! ! dx ((v ⇧ N k 1 u 0 , ! ! ! )) k N k ).
So, we deduce from the continuity property (5.2.15) that

sup ! ! ! 2X a ↵ (v , ! ! ! ) a ↵ (v , ! ! ! ) ||! ! ! || ↵  10( K X k=1 ↵ k ||v ⇧ N k 1 u 0 || 2 L 2 (⌦ k ) d ) 1 2  10||u 0 v || ↵ + 10( K X k=1 ↵ k ||u 0 ⇧ N k 1 u 0 || 2 L 2 (⌦ k ) d ) 1 2 .
Because of the well-known properties of the operator ⇧ N k 1 ([21], Theorem 7.3), we can get the following estimate: if the solution u 0 satisfies that u 0|⌦ k belongs to

H s k +1 (⌦ k ) d , s k 0 sup ! ! ! 2X a ↵ (v , ! ! ! ) a ↵ (v , ! ! ! ) ||! ! ! || ↵  4||u 0 v || ↵ + c( K X k=1 ↵ k N 2s k k ||u 0|⌦ k || 2 H s k (⌦ k ) d ) 1 2 .
(5.3.13)

Similarly, for any ! ! ! in X , we have

((↵f, ! ! ! )) ↵ K X k=1 ↵ k Z ⌦ k f • ! ! ! dx = K X k=1 ↵ k (((I f ⇧ N k 1 f, ! ! ! )) k N k Z ⌦ k (f ⇧ N k 1 f) • ! ! ! dx).
So, we also apply the exactness property of the Gauss-Lobatto formula, which yields

((↵f, ! ! ! )) ↵ K X k=1 ↵ k Z ⌦ k f • ! ! ! dx  p ↵ max (10( K X k=1 ||f ⇧ N k 1 f|| 2 L 2 (⌦ k ) d ) 1 2 +9||f I f|| L 2 (⌦ k ) d )||! ! ! || L 2 (!) d .
According to the approximation properties of the operator I ([21], Theorem 7.1) and

⇧ N k 1 ([21], Theorem 7.
3), we derive that, if the function f which satisfies that

f |⌦ k belongs to H k (⌦ k ) d , k > 1.
sup

! ! ! 2X ((↵f, ! ! ! )) ↵ K P k=1 ↵ k R ⌦ k f • ! ! ! dx ||! ! ! || ↵  c p ↵ max ( K X k=1 N 2 k k ||f |⌦ k || 2 H k (⌦ k ) d ) 1 2 . (5.3.14)
By analogy, we estimate the numerical integration term (5.3.9), we have

sup ! ! ! 2X a ↵ (u b , ! ! ! ) a ↵ (u b , ! ! ! ) ||! ! ! || ↵  c p ↵ max ( K X k=1 N 2s k k ||u b|⌦ k || 2 H s k (⌦ k ) d ) 1
2 .

(5.3.15) Also, the term (5.3.11) derives that

sup ! ! ! 2X b(! ! ! , P ) b (! ! ! , p b ) ||! ! ! || ↵  c p ↵ max ( K X k=1 N 2s k k || P|⌦ k || 2 H s k +1 (⌦ k ) d ) 1 2 . ( 5
||u b u b || ↵  c p ↵ max K 1 X k=1 N s k k ||k|| H s k 1 2 ( 1 \@⌦ k ) .
(5.3.17)

where K 1 is such that 1 \ @⌦ k 6 = ;.

We are now in the position to derive the approximation error in X (5.3.6). We refer to ( [START_REF] Daadaa | Discrétisation spectrale et par éléments spectraux des équations de Darcy[END_REF], Chap.3, Lemma. 3.24) for the following lemma.

Lemma 5.5 There exists a constant c independent of such that

inf v 2V ||u 0 v || ↵  c( inf z 2X ||u 0 z || ↵ + sup q 2M 1 R S (u 0 • n)[q ]d⌧ ⌧ ⌧ ||q || ↵ ⇤
).

(5.3.18)

Proof: Let z be an arbitrary element of X . The inf sup condition (5.2.17) and Theorem 5.3 If the assumptions of T heorem6.3 are satisfied in dimension 2. In dimension 3, it needs that the decomposition be conforming and the the following error estimate holds between the pressure p of problem (??) and the pressure p of problem (5.2.7):

||p p || ↵ ⇤  c((1 + µ + µ 0 ) K X k=1 N s k k ( p ↵ max (log N k )||u |⌦ k || H s k (⌦ k ) d + r 1 ↵ min ||p |⌦ k || H s k +1 (⌦ k ) ) + p ↵ max K X k=1 N k k ||f |⌦ k || H k (⌦ k ) d + p ↵ max K 1 X k=1 N s k k ||k|| H s k 1 2 ( 1 \@⌦ k ) + (1 + µ + µ 0 ) r 1 ↵ min K 2 X k=1 N s k k ||p 0 || H s k + 1 2 ( 2 \@⌦ k ) ).
In the two-dimensional case of a polygon ⌦, a more explicit estimate can be deduced from the previously quoted regularity results. In the end of this part, we give the following corollary according to ( [START_REF] Bernardi | Polynomial approximation of some singular functions[END_REF]) and ( [START_REF] Azaïez | The mortar spectral element method in domains of operators. part I: The divergence operator and Darcy's equations[END_REF]).

Corollary 5.2 In the two-dimensional case of a polygon ⌦, assume that the date

(f, k, p 0 ) belongs to H k (⌦ k ) 2 ⇥ H k 1 2 ( 1 \ @⌦ k ) ⇥ H k + 1 2 ( 2 \ @⌦ k ), k > 1
Then, the following error estimate holds between the solution pair (u, p)of problem (??) (2.3.9) and the solution pair (u , p ) of problem (5.2.7):

||u u || ↵ + ||p p || ↵ ⇤  c(1 + µ + µ 0 ) max{ p ↵ max , r 1 ↵ min }( K X k=1 E k (log N k ) ⌫ ||f |⌦ k || H k (⌦ k ) 2 + K 1 X k=1 E k ||k|| H k 1 2 ( 1 \@⌦ k ) + K 2 X k=1 E k ||p 0 || H k + 1 2 ( 2 \@⌦ k ) ).
where ⌫ equals to zero in the conforming decomposition and to 1 otherwise, and

E k , 1  k  K is equal to N k k if ⌦k contains no corner of ⌦, sup{N k k , N 4 3 k (log N k ) 1 2 }, if ⌦k contains a nonconvex corner of ⌦ and sup{N k k , N 4 k (log N k )

Numerical implementation

Firstly, we briefly explain how to implement the discrete problem in section before.

The full set of unknowns of the discrete system is given by

• the vector U of the values if u at all nodes (x k i , y k i ), 0  i, j  N k , 1  k  K, in dimension d = 2, (x k i , y k i , z k p ), 0  i, j, p  N k , 1  k  K, in dimension d = 3,
• the vector P of the values if p at all nodes

(x k i , y k i ), 0  i, j  N k , 1  k  K, in dimension d = 2, (x k i , y k i , z k p ), 0  i, j, p  N k , 1  k  K, in dimension d = 3.
Let (U, P ) denote the vector made of these values. Then the mortar condition can be expressed in the following way: there exists a rectangular matrix Q such that the boundary and matching conditions can be enforced, which leads to a new vector QU .

Then problem 5.2.6 5.2.7 can now equivalent to the following linear system

0 @ A BQ Q T B T 0 1 A 0 @ U P 1 A = 0 @ F Q T G 1 A , (5.4.1)
where Q T stands for the transposed matrix of Q. The matrix A is fully diagonal, its diagonal terms are the ⇢ x,k i , ⇢ y,k j or the ⇢ x,k i , ⇢ y,k j , ⇢ z,k p according to the dimension. The matrix B is only block-diagonal, with K blocks B k on the diagonal, one for each ⌦ k .

Then the system can be solved by conjugate gradient algorithm. We then consider the square ⌦ = [ 1, 3] 2 , with a nonconforming decomposition into two squares k , (log N k ) ⌫} , where ⌫ is equal to 0 if the decomposition is conforming, to 1 otherwise. We can see from the above statements that the first mortar spectral discretization derives a better (optimal) convergence result for the velocity. The second mortar spectral discretization derives a better (optimal) convergence result for the pressure. results.

⌦ 1 = [ 1, 1]

The results about the numerical compution

In our later work, we will extended to unsteady Darcy's system, which models the time-dependent flow of an incompressible fluid such as water in a rigid porous medium.

8 > > > > > > > > > < > > > > > > > > > : @ t u(x, t) + ↵u(x, t) + grad p(x, t) = ↵f(x, t) in ⌦ ⇥ [0, T ], div u(x, t) = 0 in ⌦ ⇥ [0, T ], u(x, 0) = u 0 (x, 0) in ⌦, u(x, t) • n = k(x, t) on 1 ⇥ [0, T ], p(x, t) = p 0 (x, t) on 2 ⇥ [0, T ] (6.0.1)
It also admits two equivalent variational formulations as the steady Darcy's system. For time adaptivity, in order to avoid Courant-Friedrichs-Lévy conditions, an implicit scheme such as the backward Euler's scheme could be used. Concerning the space discretization, it can be extended from the previous problems. A priori error estimates will also be proved to justify the optimal convergence properties of the discretizations.

tion for two second order differential equations. A similar approach has been applied in the Legendre spectral collocation solution of the same problem in [START_REF] Bialecki | A Legendre spectral collocation method for the biharmonic Dirichlet problem[END_REF]. In [START_REF] Eh Doha | Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials[END_REF],

the authors present some efficient spectral algorithms based on the Jacobi-Galerkin methods for fourth-order equations. Theoretical work of the two methods is referred to [START_REF] Bernardi | Some spectral approximations of two-dimensional fourth-order problems[END_REF][START_REF] Bernardi | Spectral methods[END_REF][START_REF] Funaro | Some results about the pseudospectral approximation of one-dimensional fourth-order problems[END_REF].

However, to the authors' knowledge, there is little previous work on Petrov-Galerkin methods for fourth-order differential equations. Because of the merits of Petrov-Galerkin methods, that is, the test functions do not satisfy all the boundary conditions, they are widely used in computation [START_REF] Elsayed | Efficient Chebyshev-Petrov-Galerkin method for solving second-order equations[END_REF][START_REF] Ismail | Numerical solution of complex modified Korteweg-de Vries equation by Petrov-Galerkin method[END_REF][START_REF] Ma | A Legendre-Petrov-Galerkin and Chebyshev Collocation Method for third-order differential equations[END_REF][START_REF] Ma | Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation[END_REF]. This motivated our interest in such methods.

As is pointed out in [START_REF] Gottlieb | Numerical analysis of spectral methods[END_REF][START_REF] Shen | Efficient spectral-galerkin method II. direct solvers of second-and fourth-order equations using Chebyshev polynomials[END_REF], the direct application of the tau method for fourthorder equations is numerically unstable, because it results by ill conditioned system.

Meanwhile, it is difficult to get optimal error estimate in L 2 -norm. Therefore, in this study, we present a Legendre Petrov-Galerkin (LPG) method for fourth-order equations in one dimension. Through the numerical experiments, we can note that the LPG method is an efficient algorithm. Moreover, optimal error estimates in L 2norm of the LPG method can be obtained.

In a recent work [START_REF] Shen | Optimal error estimates of the Legendre tau method for second-order differential equations[END_REF], a linear steady fourth-order equation has been computed by the LPG method and get desirable numerical results. However, the algebraic systems and error estimates are not given. In this work, we give the algebraic scheme of LPG method and prove the optimal rate of convergence in L 2 -norm for linear fourthorder differential equations. Moreover, for nonlinear equations such as the Kuramoto-Sivashinsky(K-S) equation which is an important equation in physical models, a Legendre Petrov-Galerkin and Chebyshev collocation (LPG-CC) method is developed.

The scheme of the linear part is formulated in the LPG form while the nonlinear term is treated with the Chebyshev collocation method. The time discretization is a classical leapfrog/Crank-Nicolson scheme. To get optimal error estimate, we apply the approach of [START_REF] Shen | Optimal error estimates of the Legendre tau method for second-order differential equations[END_REF] in which a norm stronger than standard L 2 -norm is used.

The chapter is organized as follows. In the next section, we propose the LPG scheme and the LPG-CC scheme for linear and nonlinear fourth-order equations, respectively. Some numerical experiments are presented in Section 3. In Section 4, we recall some recent results on some projections and a Chebyshev-Gauss-Lobatto interpolation with a negative weight. In Section 5, optimal error estimates of the two schemes are given. The final section is for concluding remarks.

Chebyshev polynomials

Classical reference on the Chebyshev polynomials are [START_REF] Gottlieb | Numerical analysis of spectral methods[END_REF]. The Chebyshev polynomials of the first kind, T k (x), k = 0, 1, ..., are the eigenfunctions of the singular Sturm-Liouville problem

( p 1 x 2 T 0 k (x)) 0 + k 2 p 1 x 2 T k (x) = 0.
For any k, T k (x) is even if k is even ,and odd if k is odd. If T k is normalized so that

T k (1) = 1, then T k (x) = cos k✓, ✓ = arccos x. ( 7
T k+1 (x) = 2xT k (x) T k 1 (x), (7.2.2)
with T 0 (x) = 1 and T 1 (x) = x. > > > < > > > :

x 0 = 1, x N = 1, ! 0 = ! N = ⇧ 2N , x j = cos ⇡j N , ! j = ⇧ N , 1  j  N 1.
Let{x i} N i=0 be the Chebyshev-Gauss-Lobatto points, i.e. x i = cos i⇧ N , and u be a function on [START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF][START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF]. The Chebyshev interpolation polynomial I C N is defined as a polynomial of degree less than or equal to N such that

I C N u(x i ) = u(x i ), i = 0, 1, ..., N. (7.2.3)
7.3 Spectral discretization scheme

Linear equation

At first, we consider the following 1 dimension linear fourth-order equation

8 < : @ 4 x U @ 2 x U + ↵U = g, x 2 ( 1, 1), U (±1) = @ x U (±1) = 0, (7.3.1)
where ↵, > 0.

Let V N = P N (I)\H 2 0 (I)\H 3 (I), W N 2 = P N 2 (I)\H 1 0 (I)\H 2 (I), where I = [ 1, 1]. A Legendre Petrov-Galerkin method for the problem (7.3.1) is to find u N 2 V N such that for any v 2 W N 2 , ↵(u N , v) (@ 2 x u N , v) (@ 3 x u N , @ x v) = (g, v). (7.3.2)
Appropriate base functions of V N and W N 2 are chosen to set up the corresponding system of equations as in [START_REF] Shen | Efficient spectral-galerkin method I. direct solvers of second-and fourthorder equations using Legendre polynomials[END_REF]. For 0  n  N 4, we let 8 < :

c n = 1/ p 2(2n + 3) 2 (2n + 5), d n = 1/(2n + 1), and 8 <
:

n (x) = c n [L n (x) 2(2n+5) 2n+7 L n+2 (x) + 2n+3 2n+7 L n+4 (x)], n (x) = d n+1 [L n (x) L n+2 (x)],
where L n (x) is the Legendre polynomial of degree n. We note that all the polynomials L n (x), 0  n  N are orthogonal, and

||L n (x)|| 2 L 2 (I) = 1 n + 1 2 . Expanding u N (x) = N 4 P n=0 ûn n (x) and taking v = m (x) in (7.3.2) lead to N 4 X n=0 ⇥ ↵( n , m ) (@ 2 x n , m ) (@ 3 x n , @ x m ) ⇤ ûn = (g, m ), 0  m  N 4. (7.3.3) Let a nm = ( n , m ), b nm = (@ 2
x n , m ), p nm = (@ 3 x n , @ x m ), then

a nm = 8 > > > > > > > > > < > > > > > > > > > : 2c n d m+1 (d n + 2d n+3 ), n= m, 2c n d m+1 d n , n = m + 2, 2c n d m+1 (2d n+3 + d n+3 d n+4 /d n+1 ), n = m 2, 2c n d m+1 d n+3 d n+4 /d n+1 , n= m 4, 0, otherwise.

Nonlinear equation

Next, we consider the Kuramoto-Sivashinsky equation

8 > > > < > > > : @ t U + @ x F (U ) + @ 2 x U + @ 4 x U = g(x, t) (x, t) 2 ( 1, 1) ⇥ (0, T ) U (±1, t) = @ x U (±1, t) = 0 t 2 (0, T ) U (x, 0) = U 0 (x) x 2 ( 1, 1). (7.3.5)
where F (z) is a smooth function of z, and T > 0

The semidiscrete LPG-CC method for the problem is to find u

N (t) 2 V N such that for any v 2 W N 2 , 8 <
:

(@ t u N , v) + (@ x I C N F (u N ), v) + (@ 2 x u N , v) (@ 3 x u N , @ x v) = (I C N g(t), v) t 2 (0, T ) u N (0) = I C N u 0 , (7.3.6)
where I C N is the Chebyshev interpolation operator at the Chebyshev-Gauss-Lobatto points.

Let ⌧ be the step size in time space and S t = {k⌧ : k = 1, 2, . . . , n t ; t = n t ⌧ }. For simplicity, denote

u t = 1 2⌧ (u(t + ⌧ ) u(t ⌧ )), ū(t) = 1 2 (u(t + ⌧ ) + u(t ⌧ )).
Using leapfrog/Crank-Nicolson scheme in time such that the linear part is treated implicitly and the nonlinear part explicitly, we get the fully discrete LPG-CC approximation to (7.3.5): to find u

N 2 V N such that for any v 2 W N 2 , 8 > > > < > > > : (u N t, v) + (@ x I C N F (u N ), v) + (@ 2 x ūN , v) (@ 3 x ūN , @ x v) = (I C N ḡ, v), t 2 S T ⌧ , (u N (⌧ ), v) = (I C N [u 0 + ⌧ @ t u(0)], v), u N (0) = I C N u 0 . (7.3.7)
Note that @ t u(0) = g(0) @ 2 x u 0 @ 4 x u 0 @ x F (u 0 ). In computation, expanding u N (x) = N 4 P n=0 ûn n (x) and taking v = m (x), the scheme of the linear part can be written in a matrix form as in (7.3.4). The nonlinear term 123 and the right-hand term can be computed by the fast Legendre transform (FLT) [START_REF] Bradley | A fast algorithm for the evaluation of legendre expansions[END_REF] between the coefficients of the Legendre series and its values at the CGL points, such as

{û n } F LT ! {u N (x j )} ! {F (u N (x j ))} F LT ! {( \ I C N F (u N )) L n },
where ( \ I C N F (u N )) L n are the Legendre expansion coefficients of I C N F (u N ).

Numerical results

We first consider linear fourth-order differential equations.

Example 7.1 The 1-D fourth-order equation @ 4 x U @ 2 x U + U = ((k⇡) 4 + (k⇡) 2 + 1) sin(k⇡x), x2 I, with a smooth exact solution U (x) = sin(k⇡x).

We compute Example 7.1 by the LPG method compared with the LT method with k=6. The maximum error is listed in Table 2. Furthermore, through the appropriate combination of base functions, the tau method also possess the property of numerical stability. + cos(1) cosh( 1) cos(x) cosh(x))/(cos(2) + cosh(2))).

This example is taken from [START_REF] Ikram | A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications[END_REF] in order to make a comparison with the Sixth-order method based on non-polynomial spline functions in [START_REF] Ikram | A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications[END_REF]. We list the maximum error in Table 3. Example 7.2 shows that our method is of the same accuracy as the Sixth-order method in [START_REF] Ikram | A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications[END_REF] through taking less points.

Example 7.3 The 1-D fourth-order equation @ 4

x U + xU = (8 + 7x + x 3 )e x , 0 < x < 1.

with an analytical solution U (x) = x(1 x)e x .

This example is also taken from [START_REF] Ikram | A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications[END_REF] in order to make a comparison with the Sixthorder method based on non-polynomial spline functions in [START_REF] Ikram | A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications[END_REF]. The maximum errors of U and its nth-order derivatives (n = 1, 2, 3, 4) by the Sixth-order method in [START_REF] Ikram | A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications[END_REF] 12 1.06e-14 We note that the maximum errors computed by our method are more accurate than by the Sixth-order method in [START_REF] Ikram | A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications[END_REF], especially for the errors of nth-order derivatives (n = 1, 2, 3, 4).

Finally, we consider a nonlinear evolution equation.

Example 7.4

The Kuramoto-Sivashisky equation @ t U + U@ x U + @ 2 x U + @ 4 x U = 0, x2 I. 19 , x 0 = 10, I = [START_REF] Costabel | Un résultat de densité pour les équations de maxwell régularisées dans un domaine lipschitzien[END_REF][START_REF] Costabel | Un résultat de densité pour les équations de maxwell régularisées dans un domaine lipschitzien[END_REF] . To see the order of accuracy,we let ⌧ decrease from 10 1 to 10 4 for large N = 256 and let N increase from 64 to 256 for small ⌧ = 10 3 . The results are given in Table 6. In Table 7, L 1 and L 2 errors are recorded at time t = 0.1; 0.3; 0.5; 0.7; 1. We compare the results with those presented in [START_REF] Uddin | A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations[END_REF] by the mesh-free collocation method based on multiquadric (MQ) with c(MQ)=2.5 advancing in time by the Crank-Nicolson scheme. For the nonlinear term, they linearize it. It is noticed that the LPG-CC method shows better accuracy than the MQ method in [START_REF] Uddin | A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations[END_REF].

Preliminaries

In this section, we state approximation properties of some projection operators and an interpolation operator, which will be needed in the proof of convergence.

Throughout this chapter, C will denote a generic positive constant.

Let I = ( 1, 1), and ! ↵, (x) = (1 x) ↵ (1 + x) be a certain weight function in the usual sense. Denote by (•, •) ! ↵, and k • k ! ↵, the inner product and the norm of the weighted space L 2 ! ↵, (I), respectively. We adopt the standard notation H ! ↵, (I) for weighted Sobolev space with norm k • k ! ↵, , and semi-norm | • | ! ↵, , . The subscript ! will be omitted whenever ! ↵, (x) ⌘ 1. Let P N (I) be the space of polynomials of degree at most N on the interval I.

The Jacobi orthogonal projection operator with the weight ! ↵, (x) is defined P ↵, N : L 2 ! ↵, (I) ! P N (I). For simplicity, we set P N = P 0,0 N . Next, we introduce the spectral projection operator P k N : H k (I) ! P N (I) [START_REF] Bernardi | Spectral methods[END_REF][START_REF] Li | Super spectral viscosity methods for nonlinear conservation laws, Chebyshev collocation methods and their applications[END_REF][START_REF] Shen | Optimal error estimates of the Legendre tau method for second-order differential equations[END_REF] defined by

P k N u = u( 1) + Z x 1 (P k 1 N 1 @ ⇠ u)d⇠, k 1,
where P 0 N = P N . The projection operator has the following properties [START_REF] Li | Super spectral viscosity methods for nonlinear conservation laws, Chebyshev collocation methods and their applications[END_REF]:

(@ l x P k N u)(1) = @ l x u(1), (@ l x P k N u)( 1) = @ l x u( 1), 0  l  k 1, (7.5.1) (@ k x (P k N u u), v) = 0, 8v 2 P N k . (7.5.2)

Here we need a stronger result, which is in negative weight. ↵(e, v) (@ 2 x e, v) (@ 3 x e, @ x v) = ↵(⌘, v) (⌘, @ 2 x v). (7.6.1)

where we have used the fact (@ 3 x ⌘, @ x v) = 0 for all v 2 W N 2 , according to (7.5.2). Because of e(±1) = @ x e(±1) = 0 by (7.5.1), we can take v = ! 1, 1 e. It is easy to derive the following equalities (@ 3 x e, @ x v) = 2|@ x v(1)| 2 + 2|@ x v( 1)| 2 + 4k@ x vk 2 + k@ 2 x vk 2 ! 1,1 .

(@ 2 x e, v) = k@ x vk 2 ! 1,1 + kvk 2 .

Therefore, we have

↵kek 2 ! 1, 1 + kvk 2 + k@ x vk 2 ! 1,1 + 2|@ x v(1)| 2 + 2|@ x v( 1)| 2 + 4k@ x vk 2 + k@ 2 x vk 2 ! 1,1  k⌘k ! 1, 1 k@ 2 x vk ! 1,1 + ↵k⌘kkvk  2 k⌘k 2 ! 1, 1 + 1 4 k@ 2 x vk 2 ! 1,1 + ↵ 2 k⌘k 2 + 4 kvk 2 ,
where F = F (u N + ũN ) F (u N ). Taking v = ! 1, 1 ũN 2 W N 2 , we have

1 2 d dt kũ N k 2 ! 1, 1 + 2|@ x v(1)| 2 + 2|@ x v( 1)| 2 + 4k@ x vk 2 + k@ 2 x vk 2 ! 1,1  kI C N F k 2 + 1 4 k@ x vk 2 + kũ N k 2 ! 1, 1 + 1 4 k@ 2 x vk 2 ! 1,1 + kgk 2 1 + 1 4 k@ x vk 2 .
Let By (7.5.4), we have

u M = max 0sT {ku N (s)k L 1 (I) + k@ x u N (s)k L 1 (I) }, C F (z 1 , z 2 ) = max
kI C N F k  kI C N F k ! 1, 1  kI C N F F k ! 1, 1 + k F k ! 1, 1  CN 1 k@ x F k + k F k ! 1, 1  CN 1 || Z 1 0
F 00 (u N (s) + ✓ũ N (s))(@ x u N (s) + ✓@ x ũN (s))ũ N (s) + F 0 (u N (s) + ✓ũ N (s))@ x ũN (s)d✓|| ! 1,

+ || Z 1 0 F 0 (u N (s) + ✓ũ N (s))ũ N (s)d✓|| ! 1, 1  C F (u M , C 1 )(N 1 ||@ x ũN || ! 1, 1 + ||ũ N || ! 1, 1 )  CC F (u M , C 1 )kũ N k ! 1, 1 so, 1 2
d dt kũ N k 2 ! 1, 1 + 2|@ x v(1)| 2 + 2|@ x v( 1)| 2 + 4k@ x vk 2 + k@ 2 x vk 2 ! 1,1  C ⇤ kũ N k 2 ! 1, 1 + 1 2 k@ x vk 2 + 1 4 k@ 2 x vk 2 ! 1,1 + kgk 2 1 .
Therefore, integrating the inequality in time leads to We can get the following theorem. The process of the proof for this theorem is similar to that of the proof in [START_REF] Shen | Optimal error estimates of the Legendre tau method for second-order differential equations[END_REF].

E
Theorem 7.2 If ⇢(T )  2C 2 1 e C ⇤ T /(N + 1) 2 , then E(t)  ⇢(t)e C ⇤ t . Now, we consider the convergence of the scheme (7.3.6). Let u ⇤ = P 3 N U , ⌘ = U u ⇤ and e = u N u ⇤ . By (7.3.5), (7.3.6), (7.5.2), we have (@ t e, v) + (@ x I C N G, v) (@ 3 x e, @ x v) = (@ t ⌘, v) + (I C N g g, v) + (@ x (F (U ) I C N F (u ⇤ )), v) (e, @ 2 x v) + (⌘, @ 2 x v) + (@ 3 x ⌘, @ x v)

= (@ t ⌘, v) + (I C N g g, v) + (@ x I C N (F (U ) F (u ⇤ )), v) + (@ x (F (U ) I C N F (U )), v) (e, @ 2 x v) + (⌘, @ 2 x v)

= I 1 + I 2 + I 3 + I 4 + I 5 + I 6 , where G = F (u ⇤ + e) F (u ⇤ ). Taking v = ! 1, 1 e, we can get

|I 1 | = |(@ t ⌘, v)|  k@ t ⌘k 2 1 + 1 4 k@ x vk 2  CN 2r k@ t U k 2 r 1 + 1 4 k@ x
|I 5 | = |(e, @ 2 x v)|  kek 2 ! 1, 1 + 1 4 k@ 2 x vk 2 ! 1,1 , |I 6 | = |(⌘, @ 2 x v)|  k⌘k 2 ! 1, 1 + 1 4 k@ 2 x vk 2 ! 1,1  CN 2r kU k 2 r + 1 4 k@ 2 x vk 2 ! 1,1 ,
For the initial error,we have from (7.5.3) and (7.5.4) that

ke(0)k ! 1, 1  k(I C N P 3 N )U 0 k ! 1, 1  CN r kU 0 k r .
Just as the proof of stability, we can get the following convergence theorem.

Theorem 7.3 Assume that r 3 and U 2 H 1 (0, T ; H r (I) \ H 2 0 (I)), F (z) 2 C r (R) and g 2 L 2 (0, T ; H r 1 (I)), then there exists a positive constant C ⇤ such that when N is large enough, we have

kU (t) u N (t)k  kU (t) u N (t)k ! 1, 1  C ⇤ N r , t2 (

Conclusion

In this work, we develop the Legendre Petrov-Galerkin method for linear fourthorder equation and the Legendre Petrov-Galerkin and Chebyshev collocation method in space combined with leapfrog/Crank-Nicolson method in time for the Kuramoto-Sivashinsky equation. We present some numerical results which demonstrate the effi-

.2. 1 )

 1 Next we are coming to the trace theorems. The trace on the boundary of a function v 2 H s (⌦) is ,in a sense to make precise, the value of v restricted to . If we denote by C 0 ( ⌦) the space of continuous functions on ⌦, the precise result reads as follows, see ([40], Chapter I): Theorem 2.1 Let ⌦ be a bounded open set of R N with Lipschitz-continuous boundary and let s > 1/2.

  1.1) by a function v in Y (⌦), integrating by parts, we obtain that the first equation of (2.3.7) is satisfied for all v in Y (⌦), hence for all v in X 0 (⌦) by density. Conversely, letting v run through D(⌦ [ 2 ) \ X 0 (⌦) gives the first line of (2.1.1) in the distribution sense and letting v run through Y (⌦) leads to the fourth line. Due to the density of D(⌦) in L 2 (⌦), the second line in (2.1.1), when taken in the distribution sense, is fully equivalent to the second equation in (2.3.7).

Proposition 3 . 4 (

 34 [START_REF] Bernardi | Spectral, spectral element and mortar element methods[END_REF], Proposition 5) For any real number s 1, the following estimate holds for any

Figure 3 - 1 :

 31 Figure 3-1: The Grid of Gauss Lobatto points when N=18 and d=2 We denote by I N the Lagrange interpolation operator on the grid ⌅ N , with values in P N (⌦). The operator coincides with i (x) N i (y) N in dimension d = 2, and with i (x) N i (y) N i (z) N in dimension d = 3. So, proving its approximation properties is now obvious.

  )

Lemma 3 . 4

 34 There exists a divergence-free function u b N in D N which satisfies

Theorem 3 . 1

 31 For any data (f, k, p 0 ) such that each f, and k, p 0 are continuous on ⌦k and on 1 , 2 , problem (3.5.4) (3.5.5) has a unique solution (u N , p N ) in D N ⇥ M N .

1 p 2 )

 12 (3.5.22) Proof: The pair(u = u 0 N +u b N , p) is a solution of problem (3.5.4)-(3.5.5), and estimate (3.5.22) is a consequence of (3.5.21) and (3.5.62). On the other hand, let (u 1 , p 1 ) and (u 2 , p 2 ) be two solutions of problem (3.5.4)-(3.5.5). Then the difference (u 1 u 2 , p is a solution of problem (3.5.10) with data u b , f, p 0 equal to zero. Thus, it follows from (3.5.22) that it is zero. So the solution of problem (3.5.4)-(3.5.5) is unique.

  .1.1), u N the solution of equation (3.5.4)-(3.5.5) and ! ! ! N any function in the kernal V N (⌦). Multiply the first line of (2.1.1) by ! ! ! N gives

  .5.[START_REF] Bernardi | A new nonconforming approach to domain decomposition: the mortar element method[END_REF] In(3.5.24), letting ! ! ! N = u 0 N v N and subtracting the first line of (3.5.10) leads to

( 3 .

 3 5.29) Next, we apply equation (2.1.1) to the function v N , integrate by parts and add it to the previous line, which yields

  3.1) or ([7], Lemma 4.1). Lemma 3.7 If p 0 is continuous on 2 , then there exists a function p b N in M N and a constant c independent of N , such that p b N = I 2 N p 0 on 2 , (3.5.37)

.5. 41 )

 41 Proof: We use the Cauchy-Schwarz inequality to (3.4.6), we can derive

. 46 )

 46 It is obvious that v N 2 X N . Then we have the above inf-sup condition. From (3.5.40-3.5.41) and the inf-sup condition (3.5.45), the saddle-point problem (3.5.39) has a unique solution (u N , p 0 N ) which satisfies

Theorem 3 . 4

 34 For any data (f, k, p 0 ) such that each f, and k, p 0 are continuous on ⌦k and on 1 , 2 respectively, problem (3.5.34) (3.5.35) has a unique solution (u N , p N ) in X N ⇥ M N . Moreover, there exists a constant c independent of N such that this solution satisfies

1 2

 1 .5.49) Thus, the function u b N = grad ' N , satisfies ||u b N || ↵  c p ↵ max ||I 1 N k|| H ( 1 ) . (3.5.50) On the other hand , it follows from the standard results on saddle-point problems ([40], Chap.I, Cor.4.1), that the problem : Find

  is a solution of problem (3.5.34) -(3.5.35), and estimate (3.5.48) follows from (3.5.50) and (3.5.52).2)The pair(u N , p N = p 0 N +p b N ) is a solution of problem (3.5.34) -(3.5.35), and estimate (3.5.48) is a consequence of (3.5.38) and (3.5.47). On the other hand, let (u N 1 , p N 1 ) and (u N 2 , p N 2 ) be two solutions of problem (3.5.34) -(3.5.35). Then the difference (u N 1 u N 2 , p N 1 p N 2 ) is a solution of problem (3.5.39) with data u b N , f, p 0 equal to zero. Thus, it follows from (3.5.48) that it is zero. So the solution of problem (3.5.34) -(3.5.35) is unique.

( 2 . 3 . 9 )

 239 and the solution (u N , p N ) of problem (3.5.34)-(3.5.35).

  .5.56) Adding (3.5.55) with ! ! ! N = u 0 N v N and substracting the first line of (3.5.35) leads to

  .5.64) and ||u b u b N || ↵ (3.5.65)

  .5.71) We first use the discrete problem(3.5.35)

1 A

 1 -2 9.42e-5 9.42e-9 4.68e-13 4.8e-15||p p N || ↵ ⇤ ||p|| ↵ ⇤4.93e-2 1.99e-5 1.86e-9 1.38e-13 3.4e-15 Table2: The relative error for the second case3.6.2 The second discretizationThe first example is as follows with analytical velocity and the pressure is also with regular property: u(x, y) = 0 @ sin(⇡x) cos(⇡y) cos(⇡x) sin(⇡y) , p(x, y) = sin(⇡(x + y)).(3.6.1)

Figure 3 - 2 : 1 A

 321 Figure 3-2: The curve of convergence in L 2 (⌦) and H 1 (⌦)

Figure 3 - 3 :

 33 Figure 3-3: The curve of convergence in L 2 (⌦) and H 1 (⌦)

5 2 x(x 2 + 1) 1 4 1 ( 3

 5113 , if 1  x  1 and y = 1 and y =

Figure 3 - 4 :Figure 3 - 5 :Figure 3 - 6 : 4 Mortar

 3435364 Figure 3-4: The curve of convergence in L 2 (⌦) and H 1 (⌦)

  2.14). Here we consider separately the case of dimension d = 2 and d = 3. For each = (N 1 , N 2 , ..., N L ) and ↵ = (↵ 1 , ↵ 2 , ..., ↵ L ), we define the parameter µ

  .1.1), u the solution of equation (4.2.6)-(4.2.7) and ! ! ! any function in the kernal V (⌦). Multiply the first line of (4.1.1) by ! ! ! gives

( 4 . 2 . 40 )

 4240 Suppose the functionf |⌦ k is in H k (⌦ k ) d , k 0,then according to the properity of the orthogonal projection operator ⇧ ⇧ ⇧ N k 1 and interpolation operator I ,

In Fig. 5 -

 5 2 the curves of the errors ||u u || ↵,L 2 (⌦) and ||p p || ↵ ⇤ ,L 2 (⌦) , for both cases as a function of N, are plotted. For the smooth solution, a linear/logarithmic scale is used and we observe that the exponential decay of the error is preserved despite the non-conforming domain decomposition.

Figure 4 - 1 :

 41 Figure 4-1: The non-matching grids for N = 22 and the velocity and pressure error curves

Figure 4 - 2 : 32 Figure 4 - 3 :

 423243 Figure 4-2: The non-matching grids for a non-conforming decomposition with N = 32

¯

  m and m \ m 0 = ;, 1  m 6 = m 0  M.(5.2.2)

  the desired result follows from the Peetre-Tartar lemma ([40], Chap.I, Theorem 2.1).

  .10) sup ! ! ! 2X b(! ! ! , P ) b (! ! ! , p b ) ||! ! ! || ↵ . (5.3.11) and ||u b u b || ↵ (5.3.12)

2 , 1 AFigure 5 - 1 : 32 5. 5 2 and

 21513252 Figure 5-1: Non-matching grids for a non-conforming decomposition with N = 32

Figure 5 - 2 :

 52 Figure 5-2: The error curves for an analytical solution

For

  the two mortar discretizations, we mainly consider Dary's equations on the same domain as follows. We consider the square ⌦ = [ 1, 3] 2 , with a nonconforming decomposition into two squares ⌦ 1 = [ 1, 1] 2 , and ⌦ 3 = [1, 3]⇥[ 1, 1] and a rectangle ⌦ 2 = [ 1, 3] ⇥ [1, 3], see Fig.5-3. For an integer N 8, we take all the N k equal to N . The mortars are chosen here as {1} ⇥ [ 1, 1] [ [ 1, 3] ⇥ {1}. The condition of the numerical experment are the same then previousely.

Figure 5 - 3 :Figure 5 - 4 :

 5354 Figure 5-3: Non-matching grids for a non-conforming decomposition with N = 32

.2. 1 )

 1 Thus, the Chebyshev polynomials are nothing but cosine functions after a change of independent variable. Due to the trigonometric relation cos(k + 1)✓ + cos(k 1)✓ = 2 cos ✓ cos(k✓), we will give the recursion relation of Chebyshev polynomials:

Example 7 . 2

 72 The 1-D fourth-order equation@ 4 x U + 4U = 1, x2 I.with an analytical solution U (x) = 0.25(1 (sin(1) sinh(1) sin(x) sinh(xx)

2 q 11

 211 with exact solutionU (x, t) = C + 15 19 r 11 19 [11tanh 3 (k(x Ct x 0 )) 9tanh(k(x Ct x 0 ))]126We compute Example 7.4 by the scheme (7.3.7) at T = 1 with C = 0.1, k = 1

4 at

 4 time t = 0.1; 0.3; 0.5; 0.7; 1 with N = 120; ⌧ = 0.001; I =[START_REF] Costabel | Un résultat de densité pour les équations de maxwell régularisées dans un domaine lipschitzien[END_REF][START_REF] Costabel | Un résultat de densité pour les équations de maxwell régularisées dans un domaine lipschitzien[END_REF]; C = 0.1; x 0

Lemma 7 . 1 ( 4 ) 7 . 6

 71476 [START_REF] Shen | Optimal error estimates of the Legendre tau method for second-order differential equations[END_REF]) If u 2 H r (I) then it holds that for all 0  l  k  r,k@ l x (P k N u u)k ! l k,l k  CN l r k@ r x uk ! r k,r k , (7.5.3) 128 Denoting by I C N the Chebyshev-Gauss-Lobatto (CGL) interpolation operator, we have Lemma 7.2 ([46]) If u 2 H r (I) and r 1, then it holds that for all 0  l  1,k@ l x (I C N u u)k ! l 1,l 1  CN l r k@ r x uk ! r 1,r 1 . (7.5.Convergence of the schemes 7.6.1 Linear equation At first, we give an optimal rate of convergence for the scheme (7.3.2). Assume that the theoretical solution U (x) is smooth enough. Let ⌘ = U u ⇤ and e = u N u ⇤ , where u ⇤ = P 3 N U . From (7.3.1) and (7.3.2), we have

|z||z 1 |+|z 2 |

 2 |@ z F (z)| + (|z 1 | + |z 2 |) max |z||z 1 |+|z 2 | |@ 2 z F (z)|. Assume kũ N (s)k L 1 (I)  C 1 , 8s 2 (0, t].

vk 2 , CN 2r kgk 2 r 1 + 1 4 k@ x vk 2 .|I 3 | 4 k@ x vk 2  4 k@ x vk 2  1 4 k@ x vk 2 ,|I 4 | 4 k@ x vk 2 

 24234242142442 |I 2 | = |(I C N g g, v)|  kI C N g gk 1 k@ x vk Let U M = max 0sT {kU (s)k L 1 (I) + k@ x U (s)k L 1 (I) }. 132 = |(I C N (F (U ) F (u ⇤ )), @ x v)|  kI C N (F (U ) F (u ⇤ ))k 2 + 1 kI C N (F (U ) F (u ⇤ )) (F (U ) F (u ⇤ ))k 2 + kF (U ) F (u ⇤ )k 2 + 1 CC F (U M , kU u ⇤ k L 1 (I))N 2r kU k 2 r + = |(F (U ) I C N F (U ), @ x v)|  kF (U ) I C N F (U )k 2 + 1 CN 2r kF (U )k 2 r + 1 4 k@ x vk 2 ,

  Assume that the partition of @⌦ into 1 and 2 is sufficiently smooth for D(⌦ [ 2 ) to be dense in X 0 (⌦). If the data (f, k, p 0 ) satisfy assumption (2.3.8), any smooth enough pair of functions (u, p) in X(⌦) ⇥ L 2 (⌦) is a solution of problem (2.3.7) if and only if it is a solution of problem (2.1.1).

.3.13) 2.4 Well-posedness of the problem 2.4.1 Case of velocity in H(div , ⌦) Proposition 2.1

  in L 2 (⌦) d ⇥ H 1 (⌦). Moreover this solution satisfies ||u|| ↵ + ||p|| ↵⇤  C(||f|| ↵ + ||p 0 ||

	Thus, the function u b = grad µ satisfies
			||u b || ↵  c p ↵ max ||k||	(H	1 2 00 ( 1 )) 0 .	(2.4.22)
	On the other hand, it follows from the standard results on saddle-point problems
	([40], Chap,I, Cor.4.1) that the problems
	follows from (2.4.17) and (2.4.18) that problem (2.4.16) has a unique solution
	(u, P 1 ) in L 2 (⌦) d ⇥ H 1	2 (⌦) and that this solution satisfies
			||u|| ↵ + ||P 1 || ↵⇤  C(||f|| ↵ + || P || ↵ ⇤ )	(2.4.19)
	We are now state the main result of this section.
	1 2 ( 1 )⇥H p ↵ max ||k|| Theorem 2.6 For any data (f, k, p 0 ) in L 2 (⌦) d ⇥H has a unique solution (u, p) H 1 2 00 ( 2 ) + (H 1 00 ( 2 ), problem (2.3.9) 1 2 2 00 ( 1 )) 0 ). (2.4.20)
	Proof: We establish successively the existence and uniqueness of the solution.
	1) It follows from the Lax-Milgram lemma, combined with Bramble-Hilbert inequality,
	that there exists a unique µ in H 1	2 (⌦) such that
			Z
	8' 2 H 1	2 (⌦),	⌦	(grad µ) • (grad ')dx =< k, ' > 1 .	(2.4.21)

  .5.69) So, according to the above estimate, we have the first approximation error for the velocity.

Theorem 3.5 Assume that the solution (u, p) of the problem (2.1.1) in H m (⌦) d ⇥ H m+1 (⌦) for m d 2; the function f belongs to H r (⌦) d for r 2; the function k on 1 in H s ( 1 ) for s > d 1 2 , and p 0 on 2 in H t (⌦) for t > d 1, we have the following estimate:

6

  Assume that the solution (u, p) of the problem (2.1.1) in H m (⌦) d ⇥ H m+1 (⌦) for m d 2; the function f belongs to H r (⌦) d for r 2; the function k on 1 in H s ( 1 ) for s > d 1 2 , and p 0 on 2 in H t (⌦) for t > d 1, we have the following estimate:

  Proposition 4.1 For any data (f, k, p 0 ) such that each f| ⌦ k , 1  k  K, and k, p 0 are continuous on ⌦k and on 1 , 2 respectively, problem (4.2.6) (4.2.7) has a unique Proof: The pair(u = u 0 + u b , p ) is a solution of problem (4.2.6) and (4.2.7), and estimate (4.2.25) is a consequence of (4.2.24) and (4.2.12). On the other hand, let (u 1 , p 1 ) and (u 2 , p 2 ) be two solutions of problem (4.2.6) and (4.2.7). Then the difference (u 1 u 2 , p 1 p 2 ) is a solution of problem (4.2.13) with data u b , f, p 0

	equal to zero. Thus, it follows from (4.2.25) that it is zero. So the solution of problem
	(4.2.6) and (4.2.7) is unique.
	H	1 2 00 ( 2 )	)	(4.2.25)

solution (u , p ) in D (⌦) ⇥ M (⌦). Moreover, there exists a constant c independent of such that this solution satisfies

||u || ↵, + ||p || ↵ ⇤ ,  c(||I 1 k|| H 1 2 ( 1 ) + ||I f|| D 0 (⌦) 0 + ||I 2 p 0 ||

Table 2

 2 Maximum error for Example 7.1From Example 7.1, we note that the Legendre Petrov-Galerkin method is a little more accurate than the Legendre Tau method although the boundary conditions which test functions satisfy of the tau method are less than that of the Petrov-Galerkin method.

	N	16	32	64	128
	LT 4.02 2.37e-5 4.35e-13 4.14e-13
	LPG 5.28 1.15e-5 3.71e-13 5.14e-14

Table 3

 3 Maximum error for Example 7.2

	N Sixth-order method in [56] N	LPG
	16	9.99e-9	8 2.58e-8
	32	1.89e-10	10 7.92e-11
	64	2.82e-12	12 2.17e-13

Table 6

 6 Error at T = 1 for Example 7.4

	⌧	N L 1 error L 2 error
	1e-1		7.21e-5	1.07e-4
	1e-2 1e-3	256	7.21e-7 7.19e-9	1.07e-6 1.07e-8
	1e-4		7.08e-11	1.07e-10
		64	5.25e-2	9.37e-2
	1e-3	128	3.17e-5	1.28e-4
		256	7.19e-9	1.07e-8

Table 7

 7 Maximum error of MQ method in[START_REF] Uddin | A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations[END_REF] and LPG-CC method for Example 7.

  ⇤ is a positive constant dependent on C F (u M , C 1 ), and letE(t) = kũ N (t)k 2 ! 1, 1 +

	Z t 0	2|@ x v(1)| 2 + 2|@ x v( 1)| 2 +	7 2	|@ x vk 2 +	3 4	k@ 2 x vk 2 ! 1,1 ds,
			Z t
	⇢(t) = kũ N (0)k 2 ! 1, 1 +	0	kgk 2	1 ds.

(t)  ⇢(t) + C ⇤ Z t 0 E(⌧ )d⌧,

131

where C

Cette thèse est dédiée aux membres de ma famille, sans lesquels je n'aurais jamais vu le jour.

= v | ⌦ l(m) • n l(m) = @ ⌧ ⌧ ⌧ | ⌦ l(m) = 0 . (4.2.30)

}, if ⌦k contains a no nonconvex corner of ⌦.

Remerciements

ken"norm defined by ||v|| ↵, = (

(4.2.10)

We then construct a lifting of the boundary conditions.

Lemma 4.1 There exists a divergence free function u b in D (⌦) which satisfies

and

where C is independent of and ↵.

We set u 0 = u u b . To go further, we consider the following problem: Find (u 0 , p ) in D 0 (⌦) ⇥ M (⌦) such that 8v 2 D 0 (⌦), a ↵ (u 0 , v ) + b (v , p ) = a ↵ (u b , v ) + L (v ), 8q 2 M (⌦), b (u 0 , q ) = 0.

(4.2.13)

We introduce the kernel

We can also write it like this:

For the sake of generality, we prove the ellipticity property on a space that contains all the V (⌦).

Lemma 4.2 There exists a constant c independent of the discretization, such that the following ellipticity property about a ↵ (•, •) holds

We next prove the inf-sup condition: First introduce two spaces associated with D (⌦)

and M (⌦):

Clearly this space only depends on the decomposition.

Next, for a positive integer L, we consider the space D DD (⌦) of functions v such that

2. v • n vanish on @⌦.

3. which are continuous on ⌦.

Note that the space only depends on the decomposition and on the integer L.

Note also that it is contained in all spaces D 0 (⌦) such that the N k are larger than L.

We intend to establish that an inf-sup condition links the spaces D D (⌦) and M D (⌦),

for an appropriate choice of L. According to the reference ( [START_REF] Faker | Inf-sup conditions for the mortar spectral element discretization of the Stokes problem[END_REF], Proposition 3.1), we have the following lemma.

Lemma 4.3 There exists a constant L only depending on the decomposition, such that for any function q in M D (⌦) satisfiing 8v 2 D D (⌦), b(v, q) = 0, then q must be zero on the whole domain ⌦.

We can then derive the inf-sup condition.

In the case of dimensition d = 2, there exists a constant c independent of such that

26)

Proof: Since u is divergence-free, there exists a stream-function in

According to the isotropic case in ( [START_REF] Bernardi | A new nonconforming approach to domain decomposition: the mortar element method[END_REF],

Appendix B) and [START_REF] Bernardi | Mortar spectral element methods for elliptic equations with discontinuous coefficients[END_REF], there exists a function :

• which is equal to on @⌦,

• which preserves the values of in all the corners of the ⌦ l ,

• such that each | ⌦ l belongs to P N l (⌦ l ) and satisfies on each k,l

Here, stands for the mortar function associated with , equal to | ⌦ l(m) on each m . • and finally which satisfies (refer to ( [START_REF] Bernardi | Mortar spectral element methods for elliptic equations with discontinuous coefficients[END_REF], Lemma 3.4))

which gives the results (4.2.26). So we only need to check that it satisfies the matching condition. Let ⌧ ⌧ ⌧ denotes the unit vector dircetly orthogonal to n l(m) , the mortar function associated with v is defined by, for 1  m  M, So, we have on each k,l which is not a mortar (recall that | ⌦ k vanishes at each endpoint of k,l )

which is the desired matching condition.

We now consider the case of dimension d = 3, which is much more complex. In order to prove the approximation results, we first introduce the mortar space C (⌦)

which is the analogue of D (⌦) for the approximations in H(curl, ⌦). It is the space of functions v such that:

• the mortar function ' being defined on each m by

the following matching condition holds on each edge k,l , 1

which is not a mortar:

Here we first consider the case of a conforming decomposition, with some restrictions on the choice of the N k .

Lemma 4.5 Assume that the function u is equal to curl ⇠ ⇠ ⇠ for a function ⇠ ⇠ ⇠ in H(curl, ⌦), and is such that each

In the case of dimension d = 3, if the decomposition is conforming and if morever

where each m = k(m),l(m) is a whole edge in dimension d = 2, face in dimension d = 3, of a subdomain ⌦ k , denoted by ⌦ k(m) . Note that the choice of this decomposition is not unique, however it is decided a priori for all the discretizations we work with.

In order to describe the discrete problem, we introduce the discretization parameter , which is here a K-tuple of positive integers

Indeed, the local discrete space on each ⌦ k is the space P N k (⌦ k ) of restrictions to ⌦ k of polynomials with degree  N k with respect to each variable. In all that follows, c stands for a generic constant which may vary from one line to the other but is always independent of . We here introduce the discrete space according to (??)-(2.3.9). For each k, 1  k  K, the discrete space of velocities X is defined by

(

We give the mortar function (q) for each piecewise regular function q: On each m , 1  m  M, the restrication of (q) to m is equal to the trace of q| ⌦ k (m) . We then define the discrete space of pressures M of functions q :

1. which belong to L 2 (⌦), 2. their restriction to each ⌦ k belongs to

3. the following matching condition holds on all subdomains ⌦ k , 1  k  K, and for all k,j of ⌦ k that are not contained in @⌦ and not a mortar,

where

) is the space of polynomials with degree  N k 2 on k,j .

We construct a lifting of the boundary condition of p 0 , we give the following lemma according to the reference ( [START_REF] Bernardi | Discrétisations variationnelles de problemes aux limites elliptiques[END_REF], Th. III.3.1) or ( [START_REF] Azaïez | Spectral discretization of Darcy's equations with pressure dependent porosity[END_REF], Lemma 4.1).

Lemma 5.2 If p 0 is continuous on 2 , then there exists a function p b in M (⌦) and a constant c independent of , such that

and

.

(5.2.13)

To go further, we set p 0 = p p b , and consider the following problem:

(5.2.14)

The form a ↵ (•, •) satisfies the following continuity and ellipticity properties

The proof is similar to the reference ( [START_REF] Daadaa | Discrétisation spectrale et par éléments spectraux des équations de Darcy[END_REF], Chap.3, Lemma 3.21).

We next come to the inf-sup condition:

(5.2.17)

Proof: We note that for any q in M 1 (⌦), we can define the function v by

It is obvious that v 2 X (⌦). Then we have the above inf-sup condition.

From (5.2.16) and the inf-sup condition (5.2.17), the saddle-point problem (5.2.14) has a unique solution (u , p 0 ) which satisfies

Now we are in the position to give the main result of this section: andk, p 0 are continuous on ⌦k and on 1 , 2 respectively, problem (5.2.6) (5.2.7) has a unique solution (u , p ) in X (⌦) ⇥ M (⌦). Moreover, there exists a constant c independent of such that this solution satisfies

20)

Proof: We establish successively the existence and uniqueness of the solution. 1) It follows from the Lax-Milgram lemma and (5.2.10),that there exists a unique ' in X such that

(5.2.21)

Thus, the function u b = grad ' , satisfies

On the other hand , it follows from the standard results on saddle-point problems ( [START_REF] Girault | Finite element methods for Navier-Stokes equations: theory and algorithms[END_REF], Chap.I, Cor.4.1), that the problem

has a unique solution (u 0 , p 0 ) which morever satisfies

Then, the pair (u , p ), To conclude, we introduce the discrete kernal

As usual, it plays a key role in the numerical analysis of problem (5.2.7).

Error estimates

This section is devoted to the error estimates, first for the velocity, second for the pressure. We intend to prove here an estimate between the solution (u, p) of problem (2.3.9) and the solution (u , p ) of problem (5.2.7).

Let ! ! ! be any function in the kernal V . Multiplying the first line of (5.1.1) by ! ! !

which implies, due to the defination of V , that for any q in M (⌦),

Next, we deduce from the ellipticity property (5.2.16), that we have for any v in V

Adding (5.3.2) with ! ! ! = u 0 v and substracting the first line of (5.2.7) leads to

(5.3.4)

It is obvious that

(5.3.5) ( [START_REF] Girault | Finite element methods for Navier-Stokes equations: theory and algorithms[END_REF], Chpt. 1, Lemma 4.1) prove that there exists a unique t 2 V ? such that b (t , q ) = b (z , q ), and ||t || ↵  1 sup

Thus, if we set v = z t , then by combing the exactness property (??), we have .3.20) This inequality and triangle inequality implies

Since z is arbitrary element of X , this implies the lemma.

So the approximation error in X (5.3.6) can be divided into two parts, the first term is the approximatino error in X , another term is the consistency error.

We first estimate the approximation error in X . Taking z equal to the orthogonal projection operator ⇧ N k 1 u 0 on each ⌦ k , we can derive the following lemma.

2 .

(5.3.21)

Next we come to the consistency error. It involves the quantity µ, defined as the

We refer to ( [START_REF] Bernardi | Mortar spectral element methods for elliptic equations with discontinuous coefficients[END_REF], Lemma 3.3) and ( [START_REF] Daadaa | Discrétisation spectrale et par éléments spectraux des équations de Darcy[END_REF], Chap 3, Lemma 3.26) for the following lemma.

Lemma 5.7 For any function u such that each

, the following estimate holds Estimating the approximation error in M 1 is more technical and more complex. We first introduce the quantity which depends on ,

(5.3.24)

The nonconforming case for d = 3 is much more complicate, so we here only consider the conforming case of d = 3 and the general case of d = 2. We refer to ( [START_REF] Bernardi | Polynomials in Sobolev spaces and application to the mortar spectral element method[END_REF], Chap.

V, Theorem 5.1) and ( [START_REF] Daadaa | Discrétisation spectrale et par éléments spectraux des équations de Darcy[END_REF], Chap.3, Lemma3.27) for the following lemma.

Lemma 5.8 Assume that the solution (u, p) of problem (??) (2.3.9) is such that each

k . For further more results, we refer to ( [START_REF] Bernardi | Polynomials in Sobolev spaces and application to the mortar spectral element method[END_REF], Chap.V).

According to the above result, now we arrive at the first theorem in this section.

Theorem 5.2 Assume that the function ↵ is constant on each ⌦ k , 1  k  K. If the solution (u, p) of problem (2.3.9) is such that its restriction to each

, and the function f is such that its restrication to each

(ii) in dimension d = 3, if the decomposition is conforming, the following error estimate holds between this solution (u, p) and the solution (u , p ) of problem (5.2.7)

where µ 0 is the same as it in (5.3.26) and the constant c is independent of the parameter and the function ↵.

So it is readily checked that the following improved estimate holds for a conforming decomposition.

Corollary 5.1 If the decomposition is conforming and if the assumptions of Theorem 4.1 are satisfied, the following error estimate holds between the solution (u, p) of problem (2.3.9) and the solution (u , p ) of problem (5.2.7)

where the constant c is independent of the parameter and the function ↵.

Estimating the error on the pressure now is easy.

From the inf sup condition (5.2.17), we derive that, for any q in M 1 ,

.3.27)

We first use the discrete problem (5.2.7)

Next, we apply equation (2.3.9) to function v , and adding it to the previous line.

This yields

According to the triangle inequality, we have

So using the same arguments as in the estimation of terms issued from numerical integration together with the above triangle inequality yields

(5.3.30)

All the terms in the right-hand side have been estimated previously except the term

Because p b is in M , so the estimate result of this term is simily to the term ||P 1 q 0 || ↵ ⇤ . We refer to ( [START_REF] Bernardi | Polynomials in Sobolev spaces and application to the mortar spectral element method[END_REF], Chap. VI, Lemma 5.1 and Lemma 5.2) for the following result.

Lemma 5.9 Suppose p 0 be a function in H 1 2 ( 2 ) such that each p 0 | 2 \@⌦ k belongs to

the following error estimate holds

where µ 0 is the same as it in (5.3.26) and the constant c is independent of the parameter and the function ↵.

Now we get the second estimate result of this section on the pressure.

Chapter 6 Conclusion and Perspectives

In the whole thesis, we are mainly interested to the spectral approximation and mortar spectral element method for the Darcy's Equations with general boundary conditions.

In Chapter 3, we have presented the spectral discretization for the Darcy's equations. The spectral method has been proven optimal in the sense that the order of convergence is only limited by the regularity of the solution. The chapter 4 and Chapter 5 are the key chapters in out thesis. We propose two different discretization relies on the mortar spectral method. Both the numerical analysis of the discretization problems are performed and numerical experiments are presented, which turn out to be in good coherency with the theoretical results. The last Chapter is an appendix, which is an exception in our thesis, because a spectral scheme have been presented for fourth-order equations. We here have developed a Legendre-Petrov-Galerkin method for linear fourth-order differnetial equations in one dimension and a Legnedre-Petrov-Galerkin and Chebyshev Collocation method for nonlinear fourth-order equations.

We propose two different discretization relies on the mortar spectral method. Both the numerical analysis of the discretization problems are performed and numerical experiments are presented, which turn out to be in good coherency with the theoretical

Chapter 7

Appendix : A Spectral Method for

4-th Differential Equations

In this chapter, we develop a Legendre Petrov-Galekin method for linear fourthorder differential equations in one dimension and a Legendre Petrov-Galerkin and Chebyshev collocation method for the nonlinear Kuramoto-Sivashinsky equation. The numerical experiments are given which demonstrate the efficient of proposed schemes.

Finally, we give the optimal rate of convergence in L 2 -norm of the methods.

Introduction

There are many applications of Galerkin methods and collocation methods in the literatures for fourth-order differential equations. In [START_REF] Shen | Efficient spectral-galerkin method I. direct solvers of second-and fourthorder equations using Legendre polynomials[END_REF][START_REF] Shen | Efficient spectral-galerkin method II. direct solvers of second-and fourth-order equations using Chebyshev polynomials[END_REF], Shen proposes a class of spectral-Galerkin methods for the direct solution of second-and fourth-order equations based on Legendre and Chebyshev polynomials. In [START_REF] Bialecki | A Legendre spectral Galerkin method for the biharmonic Dirichlet problem[END_REF], a Legendre spectral Galerkin method is presented for the solution of the biharmonic Dirichlet problem.

Their approach is based on a mixed method which gives rise to a variational formula-

We denote

The scheme (7.3.3) can be written in a matrix form as follows:

(↵A B + P )û = ĝ, û = [û 0 , û1 , . . . , ûN 4 ] T . (7.3.4)

Remark 7.1 In Table 1, we list the condition numbers of the fourth-order term of the LPG method and that of the Legendre tau (LT) method which we expand u N (x) = N 4 P n=0 ûn n (x) [START_REF] My Hussaini | Spectral methods: Fundamentals in single domains[END_REF] and take v = L m (x), 0  m  N 4. The command "cond" is used in Matlab to compute the condition numbers.

2.80e-8 1.76e-7 1.08e-6 2.36e-5 5.05e-4

According to (7.5.3), we have

Furthermore, from the inverse property of polynomials (see (3.17) of [START_REF] Li | Error analysis for solving the Korteweg-de Vries equation by a Legendre pseudo-spectral method[END_REF]), we have

which means the method also admits optimal error estimate in H 1 -norm.

We arrive at the following convergence result via the triangle inequality and Lemma

Nonlinear equation

Next, we consider the Kuramoto-Sivashinsky equation (7.3.5). We give the stability and an optimal rate of convergence for the semidiscrete scheme (7.3.6) of the LPG-CC method.

Suppose that u N and the term on the right-hand side in (7.3.6) have the error ũN and g, respectively. Then by (7.3.6) we have (@ t ũN , v) + (@ x I C N F , v) + (ũ N , @ 2 x v) (@ 3 x ũN , @ x v) = (g, v).

ciency of proposed schemes, and coincide very well with theoretical analysis. Optimal error estimates in L 2 -norm of the two methods for fourth-order differential equations are also given.

Abstract

We mainly talk about in this thesis the numerical simulation of the steady flow in a rigid porous medium which is simulated by Darcy's equations with a general boundary conditions, by spectral method. The method has been proven optimal in the sense that the order of convergence is only limited by the regularity of the solution.

One of the parameters of the system depends on the permeability of the medium and, when this one is not homogeneous, the variations of the parameter could be very high. To handle this phenomenon, we propose two different discretization relies on the mortar spectral method. Both the numerical analysis of the discretization problems are performed and numerical experiments are presented, which turn out to be in good coherency with the theoretical results. In addition, we develop a Legendre Petrov-Galekin method for linear fourth-order differential equations in one dimension and a Legendre Petrov-Galerkin and Chebyshev collocation method for the nonlinear Kuramoto-Sivashinsky equation. The numerical experiments are given which demonstrate the efficient of proposed schemes. Finally, we give the optimal rate of convergence in L 2 -norm of the methods.

Keywords: porous medium, Darcys's equations, spectral method, mortar spectral element method, piecewise continuous coefficients, fourth-order equations, Kuramoto-Sivashinsky equation, Legendre Petrov-Galekin method.