On the convergence analysis of a time dependent elliptic equation with discontinuous coe cients

The connected two-dimentional domain Ω is open and bounded with a Lipschtiz-continuous boundary ∂Ω. Let T be a xed positive real. We suppose that the function λ is positive and does not depend on time. This problemwas handled in some previous works on di erent cases [1]. The case where the function λ is not globally continuous is presented in [2–4]. The a priori and a posteriori analysis were proposed based on the nite elementmethod and the spectral discretization. For the casewhere λ is piecewise constant and such that the ratio of its maximal value to its minimal value is large enough, the discretization of the stationary problem is studied in [5] by conforming nite elements and in [6] by the mortar spectral discretization. In the present work, we consider a non stationary problem with λ is piecewise constant. We proceed to the domain decomposition in two steps. Firstly, We associate a decomposition based on the value of λ (i.e. λ is constant on each sub-domain). Secondly, eachobtained sub-domain is itself decomposedon rectangles using themortar spectral method. The later is considered as themost suitablemethod for handling nonconforming decomposition (i.e the intersection of two sub-domains is not restricted to be a corner or a whole edge of both of them) [7]. The number of sub-domains can be highly reduced thanks to the non-conformity property. We refer to [8] for a rst application of this method to discontinuous coe cient in the nite element method. The discretization in time of our problem is based on the implicit Euler method. We prove that the semidiscrete problem on time is well posed and we give a time error estimate of order one. On each sub-domain,


Introduction
This paper is devoted to the numerical analysis of the mortar spectral element discretization of the heat equation in an heterogenous medium with a variable di usion coe cient λ formulated by the problem (1), (1) The connected two-dimentional domain Ω is open and bounded with a Lipschtiz-continuous boundary ∂Ω.
Let T be a xed positive real. We suppose that the function λ is positive and does not depend on time. This problem was handled in some previous works on di erent cases [1]. The case where the function λ is not globally continuous is presented in [2][3][4]. The a priori and a posteriori analysis were proposed based on the nite element method and the spectral discretization. For the case where λ is piecewise constant and such that the ratio of its maximal value to its minimal value is large enough, the discretization of the stationary problem is studied in [5] by conforming nite elements and in [6] by the mortar spectral discretization. In the present work, we consider a non stationary problem with λ is piecewise constant. We proceed to the domain decomposition in two steps. Firstly, We associate a decomposition based on the value of λ (i.e. λ is constant on each sub-domain). Secondly, each obtained sub-domain is itself decomposed on rectangles using the mortar spectral method. The later is considered as the most suitable method for handling nonconforming decomposition (i.e the intersection of two sub-domains is not restricted to be a corner or a whole edge of both of them) [7]. The number of sub-domains can be highly reduced thanks to the non-conformity property. We refer to [8] for a rst application of this method to discontinuous coe cient in the nite element method.
The discretization in time of our problem is based on the implicit Euler method. We prove that the semidiscrete problem on time is well posed and we give a time error estimate of order one. On each sub-domain, we consider a spectral discretization which approaches the solution by high degree polynomials. Since the basis of polynomials are tensorized, the sub-domains are rectangles. Di erent degrees of polynomials are chosen on each sub-domain according to the di erent values of λ. We prove that the discrete problem is well posed and we show an optimal error estimate for a good choice of domain decomposition. An outline of the paper is as follows : • In section 2 we present the continuous problem and some regularity results.
• The section 3 is about the analysis and the error estimate of the semi-discrete problem on time.
• The mortar spectral element discretization is developed in section 4.
• In section 5, we perform the estimation of the error. • Section 6 is an annex of the proof of the error estimation since it is quite technical.

The continuous problem
We denote by x=(x,y) the generic point in R , and we suppose that there exists a nite number of sub-domain We introduce some notions to clarify the spaces of functions that depend on time. The function u(x, t), de ned on the domain Ω×] , T[, can be written as: where X is a separable Banach space. We de ne C j ( , T; X) the set of time C j classes functions with a value on X. C j ( , T; X) is a Banach space for the norm where ∂ l t u is the partial derivative of order l in time of the function u. We de ne also the spaces and H s ( , T; X) = {v ∈ L ( , T; X); ∂ k v ∈ L ( , T; X); k ≤ s}.
and H s ( , T; X) is an Hilbert space for the following scalar product: Problem (1) where < ., . > is the duality product between H (Ω) and H − (Ω). We introduce the energy norm We recall the following proposition (see [9], chap 3 for its proof).

Proposition 2. We suppose that the restriction of the function λ on each sub-domain Ω
There exists a real < s < , depending on the geometry of the domain and the quotient λ max λ min , such that for f ∈ L ( , T; H s− (Ω)), the solution u of problem (3) belongs to L ( , T; H s+ (Ω) ∩ H (Ω)) for any ≤ s ≤ s .

Remark 1.
The maximum value of the real s is bounded in the following way (see [10]) where C is a constant that depends only on the domain Ω.

The time semi discrete problem
To nd the discrete problem in time, we introduce a partition of the interval [ , T]. Let [t n− , tn] the subinterval of this partition, such that = t < t < ... < t n− < ... < t M = T where M is a positive integer. We notice τ = tn − t n− , ≤ n ≤ M the step of the partition that we suppose constant. We denote by v(., tn) = v n , ≤ n ≤ M. We de ne the function vτ which is a ne on each interval [t n− , tn] by Using Euler implicit method, the semi discrete problem is written as follows : The problem (7) has the equivalent variational formulation: Let the bilinear form a n (., .) and the linear form L n (.) de ned respectively by It is easy to prove that the bilinear form a n (., .) is continuous on the space H (Ω) × H (Ω), coercive on the space H (Ω) and that the linear form L n is continuous on the space H (Ω). So according to the Lax Milgram theorem, we deduce the following proposition.

Proposition 3. For any function f in C ( , T; H − (Ω)) and u ∈ L (Ω), problem (8) has a unique solution
If we take v = u n in problem (8), we deduce the following inequality : and by making the sum on n we conclude : Proposition 4. For f in C ( , T; H − (Ω)) and u ∈ H (Ω), the solution (u n ) ≤n≤M of problem (8) satis es the following estimation Proof. To prove the estimation (10), we have to compare the two terms According to the de nition of the function uτ de ned in (6), we have ∀x ∈ Ω where '.' is the scalar product in R and |.| its associate norm.
Given that xy ≥ − x − y , thus We deduce the rst inequality of (10) by doing the sum on k. Now using the fact that xy ≤ x + y , we conclude that By doing the sum on k and using the estimation (9) we prove the second inequality of (10).
We de ne the norm || . ||n by : The a priori error estimate is the object of the following theorem.
where c is a positive constant.
Taking t = t j in problem (3), we obtain Since then using the variational formulation (8), we deduce that for any v ∈ H (Ω), the sequence (e j ), ≤ j ≤ M is a solution of problem We remark that the error e j is the solution of problem (8) for a data function By applying the mean value theorem and (9), we obtain, for ∂ t u(., t) ∈ L ( , T, H − (Ω)), the following estimation where c is a positive constant. We conclude by using the proposition 4.

The mortar spectral element discretization
In this section we consider the function λ piecewise constant. The spectral discretization requires that the elements be rectangles, which leads us to make another partition without overlapping of the domain Ω We suppose the function λ is constant on each Ω i , ≤ i ≤ I. We remark that for any ≤ i ≤ I, there exits ≤ j ≤ I • , such that Ω i ⊂ Ω • j and I > I • . To explain this problem, we take the case where I • = . This means that Ω is composed of two heterogeneous regions (see gure 1). To handle this domain by spectral discretization, we need rectangles (I = ). However, rectangles are required for a conforming decomposition. We mean by conforming that if the intersection of two rectangles Ω i and Ω j , i ≠ j is not empty, it is necessarily equal to a corner or to a hole edge of Ω i and Ω j .
We suppose that the intersection of each boundary ∂Ω i of the sub-domain Ω i with the boundary ∂Ω of the domain Ω is a corner or a hole edge of Ω i . The skeleton of the decomposition where γm is called mortar, which is equal to a hole edge of one sub-domain Ω i that we note Ω i(m) . Let the space of the polynomial functions de ned on Ω i , with degree ≤ N i , for the two variables x and y. We de ne the mortar discrete space X δ , (δ = (N , ..., N I ) is the discretization parameter) as the space of functions u δ such that (see [7]): • u δ /Ω i , ≤ i ≤ I, belongs to the polynomial space P N i (Ω i ), • u δ vanishes on the boundary ∂Ω, • let ϕ the mortar function such that ϕ /γm = u δ /Ω i(m) /γm , for each Ω i , ≤ i ≤ I and an edge Γ of Ω i , which is not included on the boundary ∂Ω, we have the following matching condition : where P N i − (Γ) is the space of polynomials with degree ≤ (N i − ), de ned on Γ. Since Γ does not always coincide with a mortar γm , ≤ m ≤ M, this allows us to say that the discretization is not conforming (X δ is not a subspace of H (Ω)).
We remind the Gauss-Lobatto quadrature formula on the interval Λ =] − , [ : If N ≥ is an integer, let ϵ = − and ϵ N = , there exists a unique set of nodes ϵ k , ≤ k ≤ (N − ) and weights ϱ k , ≤ k ≤ N, such that : Hereinafter, We recall the following property (see [11]): We nd the value of the nodes and weights ϵ x ik and ϱ x ik (respectively ϵ y ik and ϱ y ik ) in the direction x (respectively in the direction y) by homothety and translation of the domain Ω i to the reference domain Λ . So, we have the discrete scalar product de ned as : For φ and ψ continuous on each where Let | φ | d,Ω = (φ, φ) δ the associate discrete norm. We introduce the auxiliary space and I δ the Lagrange interpolation operator de ned as : We suppose for any ≤ n ≤ M, f n is continuous on each sub-domain Ω i , ≤ i ≤ I. Then we de ne the discrete problem : Find u n δ ∈ X δ for each ≤ n ≤ M, such that The bilinear form a n δ (., .), and the linear form L n δ (.), for ≤ n ≤ M are de ned as : and Since the discretization is not conforming, we de ne the broken energy norm on X δ

Lemma 1. There exist two constants c and c independent of δ such that for all v δ in X δ , we have the following equivalence :
c min( , λ min ) Proof. From (22) we deduce that || v δ || ,Ω + τλ min We conclude (23) since are equivalent with constants c and c independent of δ (see [12]).
We prove using (17), Cauchy-Schwarz inequality and lemma 1 that the bilinear form a n δ (., .) is continuous on X δ × X δ , coercive on X δ and that the linear form L n δ (.) is continuous on X δ . Using the Lax Milgram theorem, we obtain the following result.

Error estimate
For ≤ n ≤ M, we recall that u n is the solution of problem (7). In the case where λ is piecewise constant, problem (7) is written : Multiplying the rst equation in (24) by v δ ∈ X δ and integrating by parts gives where n i is the unit normal vector to ∂Ω i . If we de ne [v δ ] the jump of v δ through the skeleton S, we obtain  (8) and || w δ || X δ and c is a positive constant independent of δ.
Proof 2. Let (v n δ ) ≤n≤M ∈ X δ . By triangular inequality u n − u n δ n ≤ u n − v n δ n + v n δ − u n δ n .
To estimate the term u n δ − v n δ n, we consider the two problems (25) and (19) for w δ ∈ X δ : By doing the di erence term by term, we obtain We remark that L n is linear and continuous on X δ which is an Hilbert space for the scalar product (., .) δ . Then, by Riesz theorem, there exists a unique element g n δ ∈ X δ such that Therefore, we conclude that z n δ = u n δ − v n δ is solution to problem (8) with a data function equal to g n δ and z δ = I δ u − v δ . Consequently using theorem 1, we have We remark that which permits to conclude (26).
Let for each mortar γm ⊂ S, ≤ m ≤ M, ζ (m) is the set of subscripts i, ≤ i ≤ I, such that ∂Ω i ∩ γm has a positive measure. By estimating each term in (26), we obtain the following result proved in section 6.

Thus β ≤ . If it is not possible, we choose the mortars and the degrees of approximation polynomials such that
which may force us to make a small change in the decomposition. So we can optimize the estimation (27) without forcing the conformity of the decomposition.

Annex
This section is devoted to the proof of theorem 2. We need to estimate each term in (26).

. Estimation of E ,j a
We pose κ j = u j − u j− . By remarking that The approximation properties of the operator Π N i − are well known (see [11], Theorem 7.3 or [13], Proposition 2.6). This permit to obtain . Estimation of E ,j a Using the exactness of the quadrature formula for a polynomial of degree ≤ N − , we write : By the triangular and Cauchy-Schwarz inequalities, we obtain : then we conclude by the properties of operator Π N i − .

. Estimation of E j f
Let Π N i − the orthogonal projection from L (Ω i ) to P N i − (Ω i ). We have by the exactness of the quadrature formula, for a polynomial of degree ≤ N i − , Using (17) in each direction, we obtain Using lemma 1 to bound w δ ,Ω by w δ X δ and the approximation properties of operator Π N i − (see [11], Theorem 7.1) and I δ (see [11], Theorem 14. 2) for f j ∈ H σ i (Ω i ); σ i > , we obtain

. Estimation of E j c
Let the operator This operator is discontinuous through the skeleton S. It means that if / and c is a positive constant independent of δ. Proof 3. Let ψ the mortar function associated to w δ ∈ X δ . By the matching condition (15), for each edge Γ of ∂Ω i , ≤ i ≤ I which is not mortar, we have : where π Γ N i − is the projection operator on P N i − (Γ). We suppose now that the decomposition is conforming, then Γ = Ω i ∩ Ω i , ≤ i ≠ i ≤ I. We know that u j /Ω i ∈ H s i + (Ω i ), we study the two following cases. i) If < s i < , we de ne the trace of λ i ∂n i u j by duality: Therefore, since π Γ N i − is the orthogonal operator from H s i − / (Ω i ) into P N i − (Γ), we conclude then for any w N i − and ψ N i − in P N i − (Γ), Thanks to the Gagliardo-Nirenberg inequality, for < α i < / .
Since λ i ≤ λ i(c) and using (30) (for t = − α i and t = + α i ), we conclude that : Therefore, we end by doing the sum on i, ≤ i ≤ I. 3) Let H i , ≤ i ≤ I the set of edges of Ω i which are not include in ∂Ω and are not mortars.

Conclusion
This work concerns the numerical analysis of the mortar spectral elements method discretization of the heat equation with a di usion coe cient λ, depending on the heterogeneity of the domain. To solve the problem of the solution singularity due to the discontinuity of λ, we use a non conform geometric decomposition of the domain. We prove an optimal error estimate that depends only on the local regularity of the solution.
The numerical validation of this result will be the subject of a forthcoming work.