The Maximum Norm Analysis of a Nonmatching Grids Method for a Class of Parabolic Equation

Motivated by the idea which has been introduced by M. Haiour and S.Boulaaras (Proc. Indian Acad. Sci. (Math. Sci.) Vol. 121,No. 4, November 2011,pp.481–493), we provide a maximum norm analysis of Euler combined with finite element Schwarz alternating method for a class of parabolic equation on with nolinear source terms two overlapping subdomains with nonmatching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a stability analysis on Euler scheme which given by our work in (App. Math. Comp., 217, 6443–6450 (2011)), we establish, on each subdomain, an optimal asymptotic behavior between the discrete Schwarz sequence and the asymptotic solution of parabolic differential equations.


Introduction
This paper deals with the error analysis in the maximum norm, in the context of the nonmatching grids method, of the following evolutionary equation: find u ∈ L 2 0, T ; where Σ is a set in R 2 × R defined as Σ = Ω × [0, T ] with T¨< +∞, where Ω is a smooth bounded domain of R 2 with boundary Γ.The function α ∈ L ∞ (Ω) is assumed to be non-negative satisfies α ≤ β, β > 0. ( The function f (•) is a nonlinear and Lipschitz functions with Lipschitz constant c and satisfying the following condition    f ∈ L 2 0, T, L 2 (Ω) ∩ C 1 0, T, H −1 (Ω) c < β. .
Schwarz method has been invented by Herman Amandus Schwarz in 1890.This method has been used to solve the stationary or evolutionary boundary value problems on domains which consists of two or more overlapping sub-domains (see [1], [9], [24], [26]).We refer to ( [1], [9]- [11]), and the references therein for the analysis of the Schwarz alternating method for elliptic obstacle problems and to the proceedings of the annual domain decomposition conference beginning with [17].For results on maximum norm error analysis of overlapping nonmatching grids methods for elliptic problems we refer, for example, to [6].
In [9], we studied the overlapping domain decomposition method combined with a finite element approximation for elliptic equation related for Laplace operator ∆, where on uniform norm of an overlapping Schwarz method on nonmatching grids has been used, where we proved that the discretization on every subdomain converges on uniform norm norm.Furthermore, a result of asymptotic behavior in uniform norm has been given.In this paper, similar to that in [9], we extend the last work for evolutionary equation with mixed boundary conditions, where we provide a maximum norm analysis of a theta scheme combined with finite element Schwarz alternating method for a linear parabolic equations on two overlapping subdomains with nonmatching grids.We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid.The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one.Under a stability analysis on the theta scheme which given by our work in [3], we establish, on each subdomain, an optimal asymptotic behavior between the discrete Schwarz sequence and the asymptotic solution of parabolic differential equations with respect the nonlinearity of the right hand side.
The outline of the paper is as follows: In section 2, we introduce some necessary notations, then we prove a full-discrete weak formulation of the presented problem using Euler time scheme combined with a finite element method.In section 3 we state a continuous alternating Schwarz sequences and define their respective finite element counterparts in the context of nonmatching overlapping grids.Section 4 is devoted to the asymptotic behavior of the method.

The discrete parabolic equation
The problem (1.1) can be reformulated into the following continuous parabolic variational equation: where a (., .) is the bilinear form defined as:

The spatial discretization
We discretize the problem (2.1) with respect to time by using Euler scheme.Therefore, we search a sequence of elements u k ∈ H 1 0 (Ω) which approaches u (t k ) , t k = k∆t, with initial data u 0 = u 0 .Thus, we have for k = 1, ..., n (2.3)

The spatial discretization
Let Ω be decomposed into triangles and τ h denote the set of all those elements h > 0 is the mesh size.We assume that the family τ h is regular and quasi-uniform.We consider the usual basis of affine functions ϕ l , l = {1, ..., m (h)} defined by ϕ l (M s ) = δ ls where M s is a vertex of the considered triangulation.We introduce the following discrete spaces V h of finite element where r h is the usual interpolation operator defined by and P 1 denotes the space of polynomials with degree at most 1.
In the sequel of the paper, we shall make use of the discrete maximum principle assumption (dmp).In other words, we shall assume that the matrices (A) ps = a ϕ p , ϕ s is M -matrices (cf.[13]).
We discretize in space the problem (2.3), i.e. that we approach the space H 1 0 by a space discretization of finite dimensional V h ⊂ H 1 0 , we get the following discrete PQVIs.
which implies (2.7) Then, the problem (2.7) can be reformulated into the following coercive discrete system of elliptic quasi-variational inequalities (EQVIs) such that (2.9)

An iterative discrete algorithm
As we have chosen before in the iterative semi-discrete algorithm u 0 h = u h0 the solution of the following full-discrete equation where g 0 is a linear and a regular function.Now we give the full following discrete algorithm where be the corresponding right-hand sides to the EQVIs.Lemma 2.1.[cf.4,6] Under the previous assumption and the dmp we have, if We shall first recall some results related to coercive quasi variational inequalities that are necessarily in proving some useful qualitative properties.
Proof.The proof of the Lemma is very similar to that in ( [7] and [10]) for free boundary problem.✷ Definition 2.2.ζ k h is said to be a subsolution for the system of EQVIs (2.8 Notation 1.Let X h be the set of discrete subsolutions.Then, we have the following theorem. Theorem 2.3.Under the discrete maximum principle, the solution of the system of EQVI (2.8) is the maximum element of X h .
Proof.We denote by ϕ + = max(ϕ, 0), ϕ − = max(−ϕ, 0).Let w h ∈ V h be a solution of the following of the full discrete system of parabolic quai variational inequalities using Euler time scheme combined with a finite element spatial approximation (cf.[3,4]) where vh = Since ṽ is a trial function, we choose ṽh = w h − v h and v h > 0. Thus that is to say w h ∈ X h .On the other hand; let z h be a subsolution, such that Maximum Norm Analysis of NGM for Parabolic Equation 163and since w h is a subsolution too, we have Under the coerciveness of the bilinear, we can get Thus, from (2.14) and (2.15) we obtain z h = w h .✷ Theorem 2.4.see [9] .Under suitable regularity of the solution of problem (1.1), there exists a constant C independent of h such that Lemma 2.5.(see [20]) Proposition 2.6.Under the previous notation, we have }. (2.17) Proof.First, putting then On the other hand, we have By using the result of lemma 2.1, we get Similarly, interchanging the roles of the couples (F k , ϕ) and ( F k , ϕ k ), we get which completes the proof.✷ Remark 2.7.Proposition 2.6 stays true for the discrete case.Proposition 2.9.Let DMP hold, we have }. (2.24) Proof.The proof is similar to that of the continuous case.✷

Schwarz Alternating Methods for parabolic equation
We decompose (Ω) in two overlapping smooth subdomain Ω 1 and Ω 2 such that Ω = Ω 1 ∪ Ω 2 , we denote by ∂Ω i the boundary of Ω i and Γ i = ∂Ω i ∩ Ω j and assume that the intersection of Γ i and Γ j ;i = j is empty.Let We associate with problem (2.8) the following system: find where and

The Continuous Schwartz Sequences
Let u 0 be an initialization in C 0 Ω ,i.e., continuous functions vanishing on ∂Ω such that Starting from u 0 = u 0 /Ω 2 , we respectively define the alternating Schwarz sequences u n+1 where where ), n ≥ 0 produced by the Schwarz alternating method converge geometrically to a solution u of the elliptic obstacle problem.More precisely, there exist k 1 , k 2 ∈ (0, 1) which depend on (Ω 1 , γ 2 ) and (Ω 2 , γ1) such that for all n ≥ 0, sup

The discrete Schwartz sequences
As we have defined before, for i = 1, 2, let τ hi be a standard regular and quasiuniform finite element triangulation in Ω i ; h i , being the mesh size.The two meshes being mutually independent Ω 1 ∩ Ω 2 , a triangle belonging to one triangulation does not necessarily belong to the other and for every w ∈ C (Ω i ), we set where π hi denote an interpolation operator on Γ 0i .Now, we define the discrete counterparts of the continuous Schwarz sequences defined in (3.4) and (3.5) .
Indeed, let u 0h be the discrete analog of u 0 , defined in (3.3), we respectively, 4. Maximum norm analysis of asymptotic behavior

Error analysis for the stationary case
We begin by introducing two discrete auxiliary sequences and prove a fundamental lemma.

Two auxiliary Schwarz sequences. For w
respectively.It is then clear that w ∞,n+1 1h and w ∞,n+1 2h are the finite element approximation of u ∞,n+1 1 and u ∞,n+1 2 defined in (4.1), (4.2), respectively.Then, as , (independent i of n).Therefore, making use of standard maximum norm estimates for linear parabolic problems, we have Maximum Norm Analysis of NGM for Parabolic Equation 167where C is a constant independent of both h and n.
Notation 4. From now on, we shall adopt the following notations: , and we set π h1 = π h2 = π h .

Iterative discrete algorithm
We give our following discrete algorithm where u k h is the solution of the problem (2.8) and the first iteration u 0 h is solution of (3.3).

Proposition 4.1. [5]Under the previous hypotheses and notations, we have the following estimate of convergence
Lemma 4.2.Let ρ = λ + c β + λ .Then, under assumption (1.2), there exists a constant C independent of both h and n such that Proof.We know from standard error estimate on uniform norm for linear problem [19] that there exists a constant C independent of h such that Let us now prove (4.6) by induction.Indeed for n = 1, using the result of Proposition 1, we have in Ω We then have to distinguish between two cases so, by multiplying (4.10) by ρ we get or that is (4.12) implies and (4.13) implies It follows that only the case (4.12) is true, that is, then So, in both cases (4.9) and (4.10), we have Similarly, we have in (4.20) So, by multiplying (4.20) by ρ we get Hence is bounded by both or that is that is, both cases (4.18) and (4.19) imply Proof.Let us give the proof for i = 1.The one for i = 2 is similar and so will be omitted.Indeed, Let δ = δ 1 δ 2 , then making use of Theorem 2 and Lemma 3, we get

Asymptotic behavior
This section is devoted to the proof of main result of the present paper, where we prove the theorem of the asymptotic behavior in L ∞ -norm for parabolic variational inequalities, where we evaluate the variation in L ∞ between u h (T ) , the discrete solution calculated at the moment T = p∆t and u ∞ , the asymptotic continuous solution of (2.1) where C is a constant independent of h and k.
Proof.We have Using the Proposition 4.

Theorem 4 . 4 .
According to the results of the Proposition 3 and the Theorem 3