Maximum Norm Analysis of an Arbitrary Number of Nonmatching Grids Method for Nonlinears Elliptic PDES

We provide a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear elliptic PDE on an arbitrary number of overlapping subdomains with nonmatching grids. We consider a domain which is the union of an arbitrary number m of overlapping subdomains where each subdomain has its own independently generated grid. The m meshes being mutually independent on the overlap regions, a triangle belonging to one triangulation does not necessarily belong to the other ones. Under the a Lipschitz assumption on the nonlinearity, we establish, on each subdomain, an optimal L∞ error estimate between the discrete Schwarz sequence and the exact solution of the PDE.


Introduction
The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains.The solution is approximated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains.The effectiveness of the Schwarz methods for various classes of nonlinear elliptic PDE problems has been demonstrated in many papers; see [1][2][3][4] and the references therein.Also the effectiveness of the Schwarz methods for these problems, especially those in fluid mechanics, has been demonstrated in many papers.See proceedings of the annual domain decomposition conference beginning with [5].
In [6,7], an optimal convergence order is obtained for nonlinear elliptic PDE, in the context of two overlapping nonmatching grids, in the sense that each subdomain has its own independent discretization by finite element method.This kind of discretization is very interesting as it can be applied to solve many practical problems which cannot be handled by global discretizations.Discretizations are earning particular attention of computational experts and engineers as they allow the choice of different mesh sizes and different orders of approximate polynomials in different subdomains according to the different properties of the solution and different requirements of the practical problems.Quite a few works on maximum norm error analysis of overlapping nonmatching grids methods for elliptic problems are known in the literature; compare and confer, for example, [8][9][10].
This paper is a continuation of previous work [6], attempting to generalize the obtained result related to convergence order also in the context of nonmatching grids in domain which consists of a union of an arbitrary number of subdomains.It is proved that the error estimate remains true also for more than two subdomains.
The proof of the main result which consists of estimating the error in the maximum norm between the continuous solution of the problem and the discrete Schwarz iterates stands on a Lipschitz continuous dependency with respect to both the boundary condition and the source term for linear elliptic equations.The optimal convergence order is then derived making use of standard finite element  ∞ error estimate for linear elliptic equations.Now, we give an outline of the paper.In Section 2 we state the continuous alternating Schwarz sequences and define their respective finite element counterparts in the context of nonmatching overlapping grids.Section 3 is devoted to the  ∞ error analysis of the method.

Preliminaries
We begin by laying down some definitions and classical results related to linear elliptic equations.

Linear Elliptic Equations.
Let Ω be a bounded polyhedral domain of R 2 or R 3 with sufficiently smooth boundary Ω.We consider the following: the bilinear form  (, V) = ∫ Ω (∇ ⋅ ∇V) ; (1) the linear form the right hand side , a regular function; (3) the space where  is a regular function defined on Ω.
The proposition below establishes a Lipschitz continuous dependency of the solution with respect to the data.
Indeed, assume that the discrete maximum principle (d.m.p) holds; that is, the matrix resulting from the finite element discretization is an M-Matrix.Then one has the following.
Proof.The proof is a direct consequence of the discrete maximum principle.
We decompose Ω into  overlapping smooth subdomains Ω  such that We denote by Ω  the boundary of Ω  and the interior boundaries by We assume that the intersection of Γ  and Γ  ,  ̸ =  is empty.Let  (  )  = {V ∈  2 (Ω  ) such that V =   on Γ  , V = 0 on Ω  ∩ Ω} .
In the sequel, we define a subsolution and a supersolution of (23) Definition 8.A smooth function   is a subsolution of (23) if where . . .

Proof. (1)
We first show by induction, that each term of the Schwarz sequences is well defined in .Indeed, for  = 0, we begin by subdomain one, since  1  1 ,  1 and  1 satisfy respectively with Lemma 7 implies that In subdomain 2, the condition of the interior boundary Γ 2 is given by On the other hand  1 2 ,  2 , and  2 satisfy, respectively, Then Lemma 7 implies that The same idea is used sequentially in the remainder of the proof related to the rest of subdomains Ω  , 3 ≤  ≤ .Indeed we obtain for  = 1, . . .,  and for  = 0 Now, let us assume that and prove that Indeed, in subdomain 1 (43) implies that Since then Lemma 7 implies that Using ( 47) and (43), we can write We can easily obtain using (48) and by adopting the same approach used in subdomain one.The result related to rest of subdomains Ω  , 3 ≤  ≤ , is obtained sequentially by similar way; that is (2) We demonstrate, by induction, that the Schwarz sequences are nondecreasing.Indeed for  = 0, we have demonstrated in (50) that, for  = 1, . . ., , Now, let us assume that and prove that Begining by subdomain one, (52) implies that Making use of (54), We get by using Lemma 7 On the other hand, (52) and (56) imply then making use of (57), We get by using Lemma 7 the following result related to the second subdomain: The same idea is sequentially applied to the rest of subdomains.Finally, we obtain (3) The sequences (   ) are bounded and nondecreasing, so they are monotone converging pointwise to   ,  = 1, . . ., .By using an elliptic regularity argument, we can see that functions   satisfy the same PDE on Ω  cf.[3,12].

The Discretization.
For  = 1, . . ., , let  ℎ  be a standard regular and quasiuniform finite element triangulation in Ω  ; ℎ  being the meshsize.The  meshes is mutually independent on Ω  ∩ Ω  , ,  = 1, . . ., , where  ̸ = ; that is a triangle belonging to one triangulation does not necessarily belong to the other ones.
We consider the following discrete spaces: and for every  ∈ (Γ  ), we set where  ℎ  denotes the interpolation operator on Γ  .
The Discrete Maximum Principle (cf.[14,15]).We assume that the respective matrices resulting from the discretiztions of problems (30) are M-matrices.Note that as the meshes ℎ  are independent over the overlapping subdomains, it is impossible to formulate a global approximate problem which would be the direct discrete counterpart of problem (13).
2.5.The Discrete Schwarz Sequences.Now, we define the discrete counterparts of the continuous Schwarz sequences defined in (30).Indeed, let  0 ℎ be the discrete analog of  0 , defined in (29) and we define for  = 1, . . ., , the discrete where Notation.From now on, we will adopt the following notations  = 1, . . ., :

𝐿 ∞ Error Analysis
3.1.The Auxiliary Schwarz Problems.This section is devoted to the proof of the main result of the present paper.To that end, we begin by introducing  discrete auxiliary problems.We define the following problems; for  = 1, . . ., ,  ℎ  ∈ It is then clear that  ℎ  is the finite element approximation of   solutions of (23).Therefore, making use of standard where  is a constant independent of ℎ.

The Main Results
Theorem 10.Let  = / < 1.Then, there exists a constant  independent of both ℎ and  such that Proof.The proof of (68) will be carried by induction and is decomposed in three principal parts where each part is devoted to one of the following situations: situation (A), situation (B), and situation (C), respectively.The three situations are defined by (73) The interval ]0, 1[ which  belongs to is divided into two subintervals as follows: and in each situation (A), (B), or (C) we deal with each subinterval separately.
Part 1.We consider situation (A).We begin the proof by the first subinterval that is For  = 0, applying (73) to subdomain one, we get We have to distinguish between two cases: (1) max {       1 − Then case 2 implies that Hence in both cases 1 and 2, we obtain Similarly applying (73) to the second subdomain, we get We have to distinguish the three following cases or or Case 1 implies that Then, We note that the two last inequalities coincide with (91) and (69), respectively.Case 2 implies that By multiplying (98) by , we get We remark that or Which implies that That is It is clear that both cases (a) and (b) are true because they coincide with (91).So there is either contradiction and case 2 is impossible or cases 2 is possible and we must have Then case 2 implies which coincides with (69).Case 3 implies By multiplying (109) by  we get So ‖ 2 −  Hence, in the three cases 1, 2, and 3 we get Equations ( 91) and (120) imply max Similarly applying the same idea for the rest of subdomain Ω  ,  = 3, . . ., , orderly, we get Now, let us assume that and prove that Equation ( 124) is obtained sequentially in the order of the numbering of subdomains.We begin by subdomain 1; indeed applying (73) to subdomain one, for iteration  + 1, we get We have to distinguish the two following cases Hence in both cases 1 and 2 we get By the same way we can obtain orderly The remainder of the proof is also by induction by addopting the same idea applied in iteration one.
Part 2. We deal with situation (B).We begin by the first subinterval that is  ∈]0, 1/2[; Indeed for  = 0, (73) implies for subdomain 1,       1 − It is clear that the two possibilities are true so either there is a contradiction and case 3 is impossible or case 3 is possible only if So case 1 is impossible and in both cases 2 and 3 we obtain Equations ( 208) and (242) imply max We adopt the same approach for the other subdomains Ω  ,  = 3, . . .,  in order, we get The remainder of the proof is by induction and is by adopting the same approach used in iteration one.The last step of the proof of this Part 2 is devoted to the subinterval  ∈ [1/2, 1[ and is also by induction.Indeed for subdomain 1 and  = 0, (73) implies We have to distinguish the two following cases or Case 1 implies So We remark that (249) coincides with (70).Case 2 implies By multiplying (250) by  we get It is clear that The remainder of the proof related to iterations  ≥ 2 is by induction by adopting the same idea used in iteration one.