Mortar spectral element discretization of Darcy’s equations in nonhomogeneous medium

Mortar spectral element discretization of Darcy’s equations in nonhomogeneous medium

Mouna Daadaa

Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05 France.

[email protected]

4 mai 2010

Abstract: We consider Darcy’s equations with piecewise continuous coefficients in a boun- ded two-dimensional domain. We propose a spectral element discretization of this problem wich relies on the mortar domain decomposition technique. We prove optimal error estimates. We also performe numerical analysis of the discrete problem and present numerical experiments.

They turn out to be in good coherency with the theoretical results.

R´esum´e: Les ´equations de Darcy mod´elisent l’´ecoulement d’un fluide visqueux incompres- sible dans un milieu poreux rigide. Un des param`etres d´epend de la perm´eabilit´e du milieu et, lorsqu’il n’est pas homog`ene, les variations de ce param`etre peuvent ˆetre extrˆemement impor- tantes. Pour traiter ce ph´enom`ene, nous proposons une discr´etisation du mod`ele par ´el´ements spectraux avec joints, l’id´ee ´etant de construire une d´ecomposition du domaine telle que la perm´eabilit´e soit constante sur chaque ´el´ement de la partition. Nous effectuons l’analyse a priori de cette discr´etisation et pr´esentons quelques exp´eriences num´eriques qui confirment les r´esultats de l’analyse.

Keywords : Mortar spectral elements, discontinuous coefficients, Darcy’s equations.

1 Introduction

This paper is devoted to the analysis of the mortar spectral element discretization of the problem introduced by Darcy [14],















αu+gradp=αf, in Ω,

div u= 0, in Ω,

u·n=g, on ∂Ω,

(1)

in a bounded two-dimensional domain Ω with a Lipschitz-continuous boundary ∂Ω, and let n denote the unit outward normal vector to Ω on∂Ω. The functionαis given with positive values.

We are interested in the case where this function is not globally continuous but only piecewise smooth and also such that the ratio of the 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.

In a first step, we consider the key situation where the function α is piecewise constant. The discretization of this problem by mortar finite element discretization is studied in [9], and opti- mal a priori and a posteriori error estimates are proven. But the idea of this paper is different : the discretization rely on a domain decomposition such that, on each subdomain, the function αis constant. To this end, the mortar element technique, introduce in [11], seems especially ap- propriate since it allows for working with noncorforming decompositions, i.e. the intersection of two subdomains is not restricted to be a corner or a whole edge of both of them. A consequence of this property, in the present situation, is that the number of subdomains in order to take into account the discontinuities of α can be highly reduced. We refer to [17, Subsec. 1.5] for a first application of this method to discontinuous coefficients in the finite element framework.

Here, on each subdomain, we consider a spectral discretization. As is well known, spectral and spectral element methods rely on the approximation by high degree polynomials and on the use of tensorized bases of polynomials. For these reasons, the basic geometries are rectangles.

Even if these methods can easily be extended to convex or curved quadrilaterals, the arguments for such an extension are very technical, so we have rather avoid them in this paper. For this reasons, the subdomains that we consider are only rectangles, and we refer the reader to [16]

for the treatment of more complex geometries for a simpler problem. It was extended [3] to the bilaplacian equation where the variational space is the standard space H²(Ω) 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 usual Sobolev spaces. We also quote [2] for an application of the mortar technique to weighted Sobolev spaces, in order to handle discontinuous boundary conditions for the NavierStokes equations.

Another advantage of the mortar method is that it allows for working with independent discre- tization parameters on the subdomains. Our idea here is to use different degrees of polynomials on these subdomains, in order to take into account the different values of α. Indeed, in prac- tical situations, even the ratio of the values of α on adjacent subdomains can be high, and the intuitive idea is to take higher degrees of polynomials in the subdomains where α is large.

We perform the numerical analysis of this discretization, in order to optimize the choice of the degrees of polynomials on each subdomains as a function of the value of α and also of the

geometry of the domain, since the geometrical singular functions issued from the non-convex corners of the domains interfere with the singularities issued from the discontinuities of α.

We also present the extension of the discretization to the case of a piecewise smooth function α : this comes either from the thermic properties of the medium where the permeability coef- ficient can depend on the density or from transformation of the geometry, for instance if the coefficients are piecewise constant on convex quadrilaterals. Since handling smooth coefficient is standard in spectral methods, the only difficulty here is to preserve the efficiency of the algorithm for solving the corresponding discrete system.

Finally, the implementation of the mortar technique mainly relies on an appropriate treatment of the matching conditions on the interfaces that we briefly describe (we refer to [4] for another way of handling these conditions). We describe some numerical experiments, which are in good coherency with the analysis and justify the choice of a domain decomposition technique and the use of different degrees of polynomials.

The outline of this paper is as follows. In Section 2, we briefly recall some properties of the continuous problem. Section 3 is devoted to the numerical analysis of the mortar spectral ele- ment discretization of the problem in the case where the functionαis piecewise constant. Error estimates between the exact and discrete solutions are established in Section 4.These results are extended to the case of piecewise smooth functions in Section 4. In Section 5, we present some numerical experiments in order to compare the present method with the spectral discretization without domain decomposition or a conforming spectral element discretization.

2 The continuous problem

Let Ω be a bounded connected open set in R², with a Lipschitz-continuous boundary ∂Ω.

Throughout the paper, we make the following assumptions on the function α : there exists a finite number of domains Ω^∗_k, 1≤k ≤K^∗, such that

• they form a partition of Ω without overlapping

Ω =∪^K_k=1^∗ Ω^∗_k and Ω^∗_k∩Ω^∗_k⁰ =∅,1≤k < k⁰ ≤K^∗, (2)

• the restriction of α to each Ω^∗_k, 1≤k ≤K^∗, is continuous on Ω^∗_k,

• the restriction of α to each Ω^∗_k, 1 ≤ k ≤ K^∗, is bounded and positive, i.e. there exist constants α^max_k and α^min_k such that

α_k^max= sup

x∈Ω^∗_k

α(x)<+∞, and α_k^min= inf

x∈Ω^∗_k

α(x)>0. (3)

We set

α_max= max

1≤k≤K^∗α^max_k and α_min = min

1≤k≤K^∗α^min_k . (4)

We define H¹²(∂Ω) as the space of traces of functions of H¹(Ω) on ∂Ω, provided with the trace norm, and H⁻¹²(∂Ω) as its dual space. As usual, L²₀(Ω) stands for the space of functions in L²(Ω) with a null integral on Ω. Finally, we consider the spaceC^∞(Ω) of infinitely differentiable functions on Ω and its subspace D(Ω) of functions with a compact support in Ω.

As now well-known (see [13, XIII.1]), system (1) admits several variational formulations. We have chosen the formulation which seems the more convenient in view of the spectral element discretization. So, we consider the variational problem

Find (u, p) in L²(Ω)²×(H¹(Ω)∩L²₀(Ω)) such that











∀v ∈L²(Ω)², a_α(u,v) +b(v, p) =

K^∗

X

k=1

Z

Ω^∗_k

α(x)f(x)·v(x)dx,

∀q ∈H¹(Ω)∩L²₀(Ω), b(u, q) = hg, qi_∂Ω,

(5)

where h., .i_∂Ω denotes the duality pairing between H⁻¹²(∂Ω) and H¹²(∂Ω), while the bilinear forms a_α(., .) and b(., .) are defined by

a_α(u,v) =

K^∗

X

k=1

Z

Ω^∗_k

α(x)u(x)·v(x)dx, b(v, q) = Z

Ω

(gradq)(x)·v(x)dx. (6) In order to optimize the constants in all that follows, we introduce the α-dependent norms

kvkα =

K^∗

X

k=1

Z

Ω^∗_k

α(x)|v(x)|²dx

!¹₂

, |q|α∗ =

K^∗

X

k=1

Z

Ω^∗_k

1

α(x)|gradq(x)|²dx

!¹₂

. (7)

The fact that the semi-norm||_α∗is a norm onH¹(Ω)∩L²₀(Ω) results from a generalized Bramble- Hilbert inequality and can easily be derived thanks to the Peetre-Tartar lemma, see [15, Chap.

I, Thm 2.1].

The well-posedness of this problem was established in [9].

Proposition 1 For any data (f, g) in L²(Ω)² ×H⁻¹²(∂Ω), problem (5) has a unique solution (u, p) in L²(Ω)²×(H¹(Ω)∩L²₀(Ω)). Moreover, this solution satisfies

kuk_α+|p|_α^∗ ≤3(√

α_maxkfk_L²_(Ω)² +kgk

H⁻¹²(∂Ω)). (8)

We are also interested with the regularity properties of this solution. We recall a result witch is proven in Prop. 2.5 of [9].

Proposition 2 There exists a real number s_α, 0< s_α < ¹₂, such that the mapping : (f, g)7−→

(u, p), where(u, p) is the solution of problem (1), is continuous from H^s(Ω)²×H^s−12(∂Ω)into H^s(Ω)²×H^s+1(Ω), pour tout s ≤sα.

Remark 3 We can exhibit a maximal value sα only limited by s_α<min{1

2, c_Ω|log (1− α_min αmax

)|}, (9)

where the constant c_Ω depends only on the geometry of Ω.

3 Analysis of the Mortar Spectral Element Discretiza- tion

Throughout this section, we work with a piecewise constant functionα. We introduce a new partition of the domain without overlapping

Ω =

K

[

k=1

Ω_k and Ω_k∩Ω_k⁰ =∅, 1≤k < k⁰ ≤K, (10) such that the function α is constant on each Ωk, 1 ≤ k ≤ K (so, each Ωk is contained in an Ω^∗_k), and also that the Ω_k, 1 ≤ k ≤ K, are rectangles. Note that K can be much larger than K^∗ in order to take into account the geometry of the discontinuities of α.

The decomposition is conforming said to be means that the intersection of two different Ωk, if not empty, is a corner or a whole edge of both of them. For simplicity, we denote by α_k the constant value of α on each Ω_k, 1≤k ≤K.

We make the further (and non restrictive) assumption that the intersection of each ∂Ωk with

∂Ω is a corner or a whole edge of Ω_k. Thus, the skeleton S of the decomposition, equal to SK

k=1∂Ω_k\∂Ω, admits a decomposition without overlapping into mortars S =

M

[

m=1

γ_m tel que γ_m∩γ_m⁰ =∅, pour m6=m⁰, (11) where each γ_m is a whole edge of one of the Ω_k, which is then denoted by Ω_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 δ, which is here a K-tuple of positive integers N_k, 1 ≤k ≤ K. Indeed, the local discrete space on each Ωk is the space P^Nk(Ωk) of restrictions to Ωk of polynomials with degree ≤Nk 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 δ.

The Γ^k,j, 1≤j ≤4 are the corners of Ωk, 1≤k ≤K.

We now introduce the discrete spaces. For each k, 1 ≤ k ≤ K, the discrete space of velocities Xδ is defined by

X^δ =

vδ∈L²(Ω)²;vδ|Ω_k ∈P^Nk(Ωk)²,1≤k ≤K . (12) According to the standard mortar element approach [11, Sec. 2] and [10], we associate with each piecewise regular function q its mortar function Φ_m(q) : On each γ_m, 1 ≤ m ≤ M, the restriction of Φ_m(q) toγ_m is equal to the trace ofq|Ω_k(m). The discrete space of pressures is the space M^δ of functionsqδ

(i) which belong to L²₀(Ω),

(ii) such that their restriction to each Ω_k, 1≤k ≤K, belongs to PNk(Ω_k),

(iii) such that the following matching condition holds on all subdomains Ωk, 1≤ k ≤K, and for all edges Γ^k,j of Ω_k that are not contained in ∂Ω,

∀ϕ∈P^Nk−2(Γ^k,j), Z

Γ^k,j

(q_δ|Ωk−Φ(q_δ))(τ)ϕ(τ)dτ = 0, (13)

where PNk−2(Γ^k,j) is the space of polynomials with degree ≤N_k−2 on Γ^k,j, andτ denotes the tangential coordinate on Γ^k,j. Note that the quantity q_δ|Ωk −Φ(q_δ) represents the jump of q_δ through Γ^k,j, where Γ^k,j is not one of the γ_m.

Note that condition (13) is obviously satisfied on all Γ^k,j which coincide with a γ_m and also that, except for some rather special decomposition, the space Mδ is not contained in H¹(Ω), which means that the discretization is not conforming.

We recall the Gauss-Lobatto formula on the interval ]−1,1[ : for each positive integer N, with the notation ξ₀^N =−1 and ξ_N^N = 1, there exists a unique set of nodes ξ_j^N, 1≤ j ≤ N −1, and weights ρ_j, 0≤j ≤N, such that

∀Φ∈P^2N−1(−1,1), Z 1

−1

Φ(ζ)dζ =

N

X

j=0

Φ(ξ_j^N)ρ^N_j . (14) The ξ_j^N are equal to the zeros of the first derivative of the Legendre polynomial of the degree N and theρ^N_j are positive. Moreover, the following positivity property holds (sse [13])

∀ϕ_N ∈PN(−1,1), kϕ_Nk²_L2(−1,1) ≤

N

X

j=0

ϕ²_N(ξ_j^N)ρ^N_j ≤3kϕ_Nk²_L2(−1,1). (15) Next, on each Ω_k, we take N equal to N_k and, by homothety and translation, we construct from the ξ_j^Nk and ρ^N_j^k, 0≤j ≤N_k, the nodes and the weights ξ^(x)_kj and ρ^(x)_kj , resp. ξ^(y)_kj and ρ^(y)_kj, in the x-direction, resp. in the y-direction (the exponent N_k is omitted for simplicity). This leads to a discrete product on all functions u and v which have continuous restrictions to all Ω_k, 1≤k ≤K :

((u,v))_δ =

K

X

k=1

((u,v))^k_N

k, (16)

with

((u,v))^k_Nk =

Nk

X

i=0 Nk

X

j=0

u(ξ_ki^(x), ξ_kj^(y))v(ξ_ki^(x), ξ_kj^(y))ρ^(x)_ki ρ^(y)_kj .

It follows from the exactness property (14) that the product ((., .))_δ coincides with the scalar product of L²(Ω) whenever the restriction of the product uv to all Ωk belong toP^2Nk−1(Ωk).

Also, we defined the global scalar product on ∂Ω ((u_δ,v_δ))^∂Ω_δ = X

{Γ^k,j⊂∂Ω}

(u_δ,v_δ)^Γ_N^k,j

k , (17)

where

(u_δ,v_δ)^∂Ω_N =

2d

X

j=1

X

x∈Ξ_N∩Γ_j

u_δ(x)v_δ(x)ρ_x, (18)

We assume that the functionsf andg has continuous restrictions to all Ω_k, 1≤k≤K and

∂Ω respectively. Then, the discrete problem reads :

Find (u_δ, p_δ)∈Xδ×Mδ such that







∀v_δ ∈Xδ, a^δ_α(u_δ,v_δ) +b_δ(v_δ, p_δ) = ((αf,v_δ))_δ,

∀q_δ ∈Mδ, b_δ(u_δ, q_δ) = ((g, q_δ))^∂Ω_δ ,

(19) where the bilinears forms a^δ_α(., .) and b_δ are defined by

a^δ_α(u_δ,v_δ) =

K

X

k=1

α_k((u_δ,v_δ))^k_N

k, b_δ(v_δ, q_δ) =

K

X

k=1

((v_δ,gradq_δ))^k_N

k. (20)

Several steps are needed for proving the well-posedness of this problem.

Lemma 4 The form a^δ_α(., .) satisfies the following properties of continuity

∀u_δ ∈Xδ, ∀v_δ ∈Xδ, a^δ_α(u_δ,v_δ)≤9ku_δk_αkv_δk_α, (21) and of ellipticity

∀u_δ ∈X^δ, a^δ_α(u_δ,u_δ)≥ ku_δk²_α. (22) Proof. Thanks to a double Cauchy-Schwarz inequality, we have

a^δ_α(u_δ,v_δ)≤a^δ_α(u_δ,u_δ)¹²a^δ_α(v_δ,v_δ)¹²,

so that it suffices to bound a^δ_α(u_δ,u_δ). Thanks to the positivity property (15), we have

K

X

k=1

αkkuδk²_L2(Ω_k)² ≤a^δ_α(uδ,uδ)≤

K

X

k=1

9αkkuδk²_L2(Ω_k)². So, the desired results.

SinceMδ is not contained in H¹(Ω), we prove that the “broken” energy norm defined by

kqkα∗ =

K

X

k=1

α⁻¹_k |q|²_H1(Ωk)

!¹₂

, (23)

is still a norm on Mδ.

Lemma 5 The quantityk.k_α∗ defined in(23) is a norm onM^δ. Moreover, there exist a constant C independent of δ such that the following property holds :

∀q ∈N(Ω)∩L²₀(Ω),

K

X

k=1

kqk²_L2(Ωk) ≤C√

α_maxkqk²_α∗. (24) We suppose that N_K ≥N_D−2, when N_D denote the maximal number of the set of all vertices of Ω_k that are inside en edge of another subdomains.

For the proof see [10].

From now on, we work with the normk.kα∗, and we suppose thatN_K ≥N_D−2 is checked.

The following continuity property is obvious :

∀vδ∈X^δ,∀qδ∈M^δ, bδ(vδ, qδ)≤ kvδkαkqδkα^∗. (25) Moreover, we note that, for any q_δ in M^δ, the functionv_δ defined by

v_δ|Ωk =α_k⁻¹grad(q_δ|Ωk), ∀1≤k ≤K, (26) belongs to Xδ. So, the following inf-sup condition is derived by taking v_δ as in (26).

Lemma 6 The form b_δ(,) satisfies the inf-sup condition

∀q_δ ∈Mδ, sup

vδ∈Xδ

b_δ(v_δ, q_δ) kvδkα

≥ kq_δkα∗. (27) We introduce the Lagrange interpolation operator I_δ^k, 1 ≤ k ≤ K, operator on all nodes (ξ_ki^(x), ξ_kj^(y)), 0≤i, j ≤N_k, with values in PNk(Ω_k), and finally the global operator I_δ by

(I_δv)_|Ωk =I_δ^kv_|Ωk, 1≤k ≤K. (28) We are now in position to prove the well-posedness of problem (5).

Theorem 7 For any data (f, g) such that each f_|Ωk, 1 ≤ k ≤ K, and g are continuous on Ω_k and on ∂Ω respectively, problem (19) has a unique solution (u_δ, p_δ) in Xδ×Mδ. Moreover, there exists a constant c independent of δ suvh that this solution satisfies

ku_δk_α+kp_δkα∗ ≤c√

α_max(kI_δfk_L²_(Ω)² +kI_δ^∂Ωgk

H⁻¹²(∂Ω)), (29) Proof. We establish successively the existence and uniqueness of the solution.

1) It follows from the Lax-Milgram lemma, combined with Bramble-Hilbert inequality and the lemma 5, that there exists a unique ϕ_δ inMδ such that

∀ψ_δ ∈Mδ, ((gradϕ_δ,gradψ_δ))_δ= ((g, ψ_δ))^∂Ω_δ , Thus, the function u^b_δ =gradϕδ, satisfies

ku^b_δkα ≤ckI_δ^∂Ωgk

H⁻¹²(∂Ω). (30)

On the other hand, it follows for the standard results on saddle-point problems, see [15, Chap.

I, Cor. 4.1], combined with (22), (27) and the inf-sup condition (27), that the problem Find (u⁰_δ, p_δ)∈Xδ×Mδ such that







∀v_δ ∈Xδ, a^δ_α(u⁰_δ,v_δ) +b_δ(v_δ, p_δ) = PK^∗

k=1α_k((f,v_δ))_δ−a^δ_α(u^b_δ,v_δ),

∀q_δ ∈Mδ, b_δ(u⁰_δ, q_δ) = 0,

(31)

has a unique solution (u⁰_δ, p_δ) which moreover satisfies ku⁰_δk_α+kp_δk_α^∗ ≤c√

α_max(kI_δfk_L2(Ω)^d +ku^b_δk_α). (32) Then, the pair (u_δ, p_δ), with u_δ = u⁰_δ +u^b_δ, is a solution of problem (19), and estimate (29) follows from (30) and (32).

2) Let (uδ, pδ) be a solution of problem (19) with data (f, g) equal to zero. Takingvδ equal to u_δ in (19) and using (22) yields that u_δ is zero. Then, the fact thatp_δ is zero follows from (27).

This proves the uniqueness of the solution (u_δ, p_δ).

To conclude, we introduce the discrete kernel

V_δ ={v_δ ∈Xδ;∀q_δ ∈Mδ, b_δ(v_δ, q_δ) = 0}. (33) As usual, it plays a key role in the numerical analysis of problem (19).

4 Error estimates

This section is devoted to the proof of an error estimate, first for the velocity, second for the pressure. We intend to prove an error estimate between the solution (u, p) of problem (5) and the solution (uδ, pδ) of problem (19). So we announce the following theorem and we describe their proof.

Theorem 8 Assume that the function α is constant on each Ω_k, 1 ≤ k ≤ K. If the solution (u, p) of problem (5) is such its restriction to each Ω_k, 1 ≤ k ≤ K, belongs to H^sk(Ω_k)² × H^sk+1(Ω_k), s_k ≥ ¹₂, and if the function f is such that its restriction to each Ω_k, 1 ≤ k ≤ K, belongs to H^σk(Ω_k), σ_k>1, the following error estimate holds between this solution (u, p) and the solution (u_δ, p_δ) of problem (19)

ku−u_δk_α

≤c (1 +µ+µ_δ)¹²

K

X

k=1

N_k^−sk α

1 2

k(logN_k)ku_|Ωkk_Hsk(Ωk)² +α⁻

1 2

k kp_|Ωkk_Hsk+1(Ωk)

+√

α_maxX^K

k=1

N_k^−2σkkf_|Ω

kk²_Hσk(Ω

k)²

1/2!

(34) where the constant c is independent of the parameter δ and the function α.

Proof. For a proof we need a several steps and lemmas. Let w_δ be any function in the kernel V_δ. Multiplying the first line of (19) by w_δ gives

K

X

k=1

α_k Z

Ωk

u₀·w_δdx+b(w_δ, p) =

K

X

k=1

α_k Z

Ωk

f ·w_δdx−a_α(u_b,w_δ), using the definition of V_δ thus implies, for any q_δ inMδ,

K

X

k=1

α_k Z

Ωk

u₀·w_δdx+b(w_δ, p−q_δ) =

K

X

k=1

α_k Z

Ωk

f ·w_δdx−a_α(u_b,w_δ). (35)

So, we deduce from ellipticity property (22), that we have for any v_δ inV_δ ku⁰_δ−v_δk²_α ≤ a^δ_α(u⁰_δ −v_δ,u⁰_δ−v_δ).

Adding (35) with w_δ=u⁰_δ−v_δ and subtracting the first line of (19) leads to ku⁰_δ−v_δk²_α≤

K

X

k=1

α_k Z

Ωk

(u₀−v_δ)(x)·(u⁰_δ−v_δ)(x)dx

+

K

X

k=1

α_k Z

Ω_k

v_δ(x)·(u⁰_δ−v_δ)(x)dx−((αv_δ,u⁰_δ −v_δ))_δ +

Z

Ω

(u⁰_δ−v_δ)(x)·grad(p−q_δ)(x)dx + ((αf,u⁰_δ−v_δ))_δ−

K

X

k=1

α_k Z

Ω_k

f(x)·(u⁰_δ−v_δ)(x)dx,

−aδ(u^b_δ,u⁰_δ−vδ) +aα(ub,u⁰_δ−vδ).

By combining property of continuity (25) and triangle inequality, we derive that the error ku−u_δk_α is bounded, up to a multiplicative constant, by the sum of five terms :

• the approximation error in Xδ

vinfδ∈Vδ

ku₀−v_δk_α, (36)

• the error approximation in M^δ

inf

qδ∈Mδ

kp−q_δk_α∗, (37)

• three terms issued from numerical integration sup

wδ∈Xδ

(a_α−a^δ_α)(v_δ,w_δ)

kw_δk_α , (38)

sup

wδ∈Xδ

a_α(u_b,w_δ)−a^δ_α(u^b_δ,w_δ)

kw_δk_α , (39)

and

sup

wδ∈Xδ

((αf,w_δ))_α−PK k=1α_kR

Ωkf(x)·w_δ(x)dx

kw_δk_α . (40)

Estimating the terms issued from numerical integration is easy since they can be evaluated separately on each Ω_k. For each k, 1 ≤ k ≤ K, let Π_Nk−1 denote the orthogonal projection operator from L²(Ω_k) onto PNk−1(Ω_k). For any w_δ inXδ, since each product of Π_Nk−1u byw_δ belongs to P^2Nk−1(Ω), it follows from the exactness property (14) that

(a_α−a^δ_α)(v_δ,w_δ) =

K

X

k=1

α_k Z

Ω_k

(v_δ−Π_Nk−1u₀)·w_δdx−((v_δ−Π_Nk−1u₀,w_δ))^k_N

k

.

So, we deduce from the continuity property (21) that sup

w_δ∈Xδ

(a_α−a^δ_α)(v_δ,w_δ)

kw_δk_α ≤ 10

K

X

k=1

αkkvδ−ΠN_k−1u0k²_L2(Ωk)^d

!¹₂

≤ 10ku₀−v_δk_α+ 10

K

X

k=1

α_kku₀−Π_Nk−1u₀k²_L2(Ωk)^d

!¹₂ . The approximation properties of the operator Π_Nk−1 are well known (see, for example, Theorem 7.3 of [7] and Proposition 2.6 of [8]), they lead to the following estimate : if the solution u₀ is such that each u0|Ω_k belongs to H^sk+1(Ωk)², sk ≥0

sup

wδ∈Xδ

(aα−a^δ_α)(vδ,wδ)

kw_δk_α ≤4ku₀−v_δk_α+c

K

X

k=1

α_kN_k^−2skku0|Ω_kk²_Hsk(Ω_k)²

!¹₂

. (41)

Similarly, for any w_δ in Xδ, we have ((αf,wδ))δ−

K

X

k=1

αk

Z

Ωk

f(x)·wδ(x)dx

=

K

X

k=1

α_k

((I_δf −Π_Nk−1f,w_δ))^k_N

k− Z

Ωk

(f −Π_Nk−1f)(x)·w_δ(x)dx

.

So, using (14) yields ((αf,w_δ))_δ−

K

X

k=1

α_k Z

Ω_k

f(x)·w_δ(x)dx

≤√ αmax



10

K

X

k=1

kf −ΠN_k−1fk²_L2(Ω_k)²

!¹₂

+ 9kf − Iδfk_L²_(Ωk)²



kwδk_L²_(Ω)².

To bound kw_δk_L²_(Ω)² as a function ofkw_δk_α and the approximation properties of the operators Iδ et ΠN_k−1 (Theorem 7.1 of [7] and Theorem 14.2 of [8]), we derive that, if the function f is such that each f_|Ω

k belongs toH^σk(Ω_k)², σ_k>1, sup

wδ∈Xδ

((αf,w_δ))_δ−PK k=1α_kR

Ωkf(x)·w_δ(x)dx kw_δk_α ≤c√

α_max

K

X

k=1

N_k^−2σkkf_|Ωkk²_Hσk(Ωk)²

!¹₂ . (42) By analogy, we estimate the term (39), and by (30), we have

sup

wδ∈Xδ

a_α(u_b,w_δ)−a^δ_α(u^b_δ,w_δ) kwδkα

≤c√ α_max

K

X

k=1

N_k^−2skkub|Ω_kk²_Hsk(Ω

k)²

!¹₂

. (43)

To estimate the term (36), we need a lemma.

Lemma 9 There exists a constant c independent of δ such that

vinf_δ∈V_δku₀−v_δk_α ≤c

z_δinf∈Xδ

ku₀−z_δk_α+ sup

qδ∈Mδ

R

S(u₀·n)[q_δ]dτ kq_δk_α∗

. (44)

Proof. Let zδ be an arbitrary element of X^δ. The inf-sup condition (27) and [15] prove there exists a unique t_δ ∈V_δ^⊥ such that

b_δ(t_δ, q_δ) = b_δ(z_δ, q_δ) and kt_δk_α ≤ 1 β sup

qδ∈Mδ

b_δ(z_δ, q_δ) kq_δkα∗

.

Thus, if we set v_δ =z_δ−t_δ, then by combining the exactness property (14) and the integration by parties, we have

b_δ(u₀, q_δ) =

K

X

k=1

Z

Ωk

u₀·gradq_δdx=

K

X

k=1

Z

∂Ωk

(u₀·n)q_δdτ = Z

S

(u₀·n)[q_δ]dτ, therefore

kt_δk_α ≤C

sup

qδ∈Mδ

bδ(u0−zδ, qδ)−R

S(u0 ·n)[qδ]dτ kq_δkα∗

. This inequality and triangle inequality implies

ku₀−v_δk_α ≤ ku₀ −z_δk_α+kt_δk_α

≤c

ku₀ −z_δk_α+ sup

qδ∈Mδ

R

S(u₀·n)[q_δ]dτ kq_δk_α∗

. As zδ is arbitrary, this implies (44).

Estimating the approximation error in Xδ is derived simply by taking w_δ equal to the orthogonal projection operator Π_Nk−1u₀ on each Ω_k.

Lemma 10 Assume that the solution (u, p) of problem (5) is such that each u|Ω_k belongs to H^sk(Ω_k)² for a real number s_k, s_k >0. The following estimate holds

inf

zδ∈Xδ

ku₀−z_δk_α ≤c

K

X

k=1

α_kN_k^−2skku_|Ωkk²_Hsk(Ω

k)²

¹₂

. (45)

Now, we evaluate the consistency error. It involves the quantity µ, defined as the largest ratio

µ= max

1≤m≤M max

`∈E(m)

α⁻¹_` α⁻¹_k(m)

!¹₂

, (46)

where, for eachm, 1≤m≤M,E(m) is the set of indicesk, 1≤k ≤K, such that∂Ω_k∩γ_m has a positive measure. Note that this constant depends on the decomposition and on the choice of the mortars but not on the discretization parameter.

In order to evaluate the consistency error, we introduce the orthogonal projection operator π_N^Γk,j

k−2 from L²(Γ^k,j) onto PNk−2(Γ^k,j). We recall the following properties of this operator (see [11]) : For any nonnegative real numbers s and t, and for any functionϕ inH^s(Γ^k,j),

kϕ−π_N^Γk,j

k−2ϕk_H^−t_(Γ^k,j₎≤cN_k^s+tkϕk_H^s_(Γ^k,j₎. (47)

Lemma 11 Assume that the solution(u, p) of problem(5) is such that eachu|Ω_k, 1≤k ≤K, belongs to H^sk(Ω_k)² for a real number s_k, s_k ≥ ¹₂, the following estimate holds

sup

q_δ∈Mδ

R

S(u₀·n)[q_δ]dτ kq_δkα∗

≤c(1 +µ)

K

X

k=1

αkN_k^−2sk(logNk)ku|Ω_kk²_Hsk(Ωk)^d

!^1/2

. (48)

Remark 12 In fact, the (logN_k)¹² in (48) disappears when all the edges of ∂Ω_k which are not mortars are contained either in ∂Ω or in one mortar, however it is negligible in comparison with the N_k^sk when N_k is large enough.

Estimating the approximation error in Mδ is more complex. See [11] for the proof.

The lemma gives a bound for the approximation error. Here, we introduce the quantity

µ_δ = max

1≤m≤M max

`∈E(m)

α⁻¹_{`</sub> N<sub>`}⁻¹ α⁻¹_k(m)N_k(m)⁻¹

!¹₂

, (49)

which now depends on δ.

Lemma 13 Assume that the solution (u, p) of problem (5) is such that eachp|Ω_k, 1≤k ≤K, belongs to H^sk+1(Ω_k) for a real number s_k, s_k >0. The following estimate holds

qδinf∈Mδ

kp−qδkα∗ ≤c(1 +µ+µδ)

K

X

k=1

α⁻

1 2

k N_k^−2skkp|Ω_kk²_Hsk+1(Ω_k)

!¹₂

. (50)

For the proof we refer to [12].

From the previous remarks, and the reference [5], the following improved estimate holds for a conforming decomposition.

Corollary 14 If the decomposition (10) is conforming and if the assumptions of Theorem 8 are satisfied, the following error estimate holds between the solution (u, p) of problem (5) and the solution (uδ, pδ) of problem (19)

ku−u_δk_α

≤c (1 +µ)¹²

K

X

k=1

N_k^−sk α

1 2

kku|Ω_kk_Hsk(Ωk)² +α⁻

1 2

k kp|Ω_kk_Hsk+1(Ωk)

+√ αmax

X^K

k=1

N_k^−2σkkf_|Ωkk²_Hσk(Ω_k)²

1/2!

, (51)

where the constant c is independent of the parameter δ and the function α.

Estimate (51) is fully optimal, at least for a geometrically conforming decomposition, and the possibly high ratios between the different values of the α_k are correctly taken into account by the weighted norms.

Also, the constant√

α_max seems unavoidable, however this is negligible since the data are most often much more regular than the solution due to the discontinuity of α.

Estimating the error on the pressure is now easy.

Theorem 15 If the assumptions of Theorem 8 are satisfied, the following error estimate holds between the pressure p of problem (5) and the pressure p_δ of problem (19) :

kp−pδkα^∗

≤c (1 +µ+µ_δ)¹² K

X

k=1

N_k^−sk α

1 2

k(logN_k)ku|Ω_kk_Hsk(Ωk)² +α⁻

1 2

k kp|Ω_kk_Hsk+1(Ωk)

+√ α_max

X^K

k=1

N_k^−2σkkf_|Ωkk²_Hσk(Ω

k)²

¹₂

+X^K

k=1

N_k^−2τ

4

X

j=1

kgk²_Hτ(∂Ω)

1/2! , (52) where the constant c is independent of the parameter δ and the function α.

Proof. From the inf-sup condition (27), we derive that, for any q_δ in M^δ, βkp_δ−q_δkα∗ ≤ sup

vδ∈Xδ

b_δ(v_δ, p_δ−q_δ)

kv_δk_α . (53)

we first use the discrete problem (19) :

b_δ(v_δ, p_δ−q_δ) = ((αf,v_δ))_δ+a_δ(u_δ,v_δ)−b_δ(v_δ, q_δ).

Next, we apply equation (5) to the function v_δ, integrate by parts and add it to the previous line. This yields

b_δ(v_δ, p_δ−q_δ) =

K

X

k=1

α_k Z

Ω_k

(u−u_δ)(x)·v_δ(x)dx

+

K

X

k=1

α_k Z

Ωk

u_δ(x)·v_δ(x)dx−((αu_δ,v_δ))_δ +

Z

Ω

v_δ(x)·grad(p−q_δ)(x)dx + ((αf,v_δ))_δ−

K

X

k=1

α_k Z

Ωk

f(x)·v_δ(x)dx

+b(v_δ, q_δ)−b_δ(v_δ, q_δ) (54)

Using the same arguments as in the estimation of terms issued from numerical integration together with a triangle inequality yields

kp−q_δk_α^∗ ≤c

ku−u_δk_α+ sup

vδ∈Xδ

(aα−a^δ_α)(uδ,vδ) kv_δk_α +kp−q_δk_α∗+ sup

v_δ∈Xδ

((αf,v_δ))_δ−PK k=1α_kR

Ωkf(x)·v_δ(x)dx kv_δk_α

+ sup

vδ∈Xδ

b(v_δ, q_δ)−b_δ(v_δ, q_δ) kv_δk_α

. (55) All the terms in the right-hand side have been estimated previously.

A more explicit estimate can be deduced from the previously quoted regularity results. We refer to [6] for proof.

Corollary 16 Assume the datum f such that each f_|Ωk, 1lek ≤ K, belongs to H^σk(Ωk)², σ_k > 1, and the datum g belong to H^τ(∂Ω), τ > ¹₂. Then, the following error estimate holds between the solution (u, p) of problem (5) and the solution (u_δ, p_δ) of problem (19) :

ku−u_δk_α+kp−p_δk_α^∗ ≤cE_k

K

X

k=1

(kf_|Ω

kk_Hσk(Ωk)² +

4

X

j=1

kgk_H^τ_(∂Ω)), (56) with

E_k =















sup{N_k⁻⁴(logN_k)³², N_k^−σk}, if Ω_k contains a corner but no nonconvex corner of Ω, sup{N⁻

4 3

k (logNk)¹², N_k^−σk}, if Ωk contains a nonconvex corner of Ω, N_k^−σk, if Ωk contains no corner of Ω.