Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic PDES

We provide a maximum norm analysis of a ﬁnite element Schwarz alternating method for a nonlinear elliptic PDE 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 Lipschitz asssumption 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 results from solving a sequence of elliptic boundary value problems in each of the subdomains.
Extensive analysis of Schwarz alternating method for nonlinear elliptic boundary value problems can be found in 1-4 and the references therein . Also the effectiveness of 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 this paper, we are interested in the error analysis in the maximum norm for a class of nonlinear elliptic problems in the context of overlapping nonmatching grids: we consider a domain which is the union of two overlapping subdomains where each subdomain has its own triangulation. This kind of discretizations are very interesting as they can be applied 2 Journal of Applied Mathematics to solve many practical problems which cannot be handled by global discretizations. They 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 cf., e.g., 6-9 . To prove the main result of this paper, we proceed as in 7 . More precisely, we develop an approach which combines a geometrical convergence result due to Lions 2 and a lemma which consists of estimating the error in the maximum norm between the continuous and discrete Schwarz iterates. The optimal convergence order is then derived making use of standard finite element L ∞ -error estimate for linear elliptic equations.
In the present paper, the proof of this lemma stands on a Lipschitz continuous dependency with respect to both the boundary condition and the source term for linear elliptic equations see Proposition 2.7 while in 7 the proof stands on a Lipschitz continuous dependency only with respect to the boundary condition for the elliptic obstacle problem.
To the best of our knowledge, this paper provides the first L ∞ -error analysis for overlapping nonmatching grids for nonlinear elliptic PDEs. We also believe that our convergence result will have important implications in the computation of the solution of this type of problems on composite grids. Now, we give an outline of the paper. In Section 2 we state a continuous alternating Schwarz sequences and define their respective finite element counterparts in the context of nonmatching overlapping grids. Section 3 is devoted to the L ∞ -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 bilinear form where g is a regular function defined on ∂Ω.
We consider the linear elliptic equation: find ξ ∈ V g such that where c is a positive constant such that Let V h be the space of finite elements consisting of continuous piecewise linear functions v vanishing on ∂Ω and ϕ s , s 1, 2, . . . , m h be the basis functions of V h .
The discrete counterpart of 2.5 consists of finding and π h is an interpolation operator on ∂Ω.

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Proof. First, set Then

2.13
On the other hand, we have

2.15
Thus, making use of Lemma 2.2, we get Similarly, interchanging the roles of the couples f, g and f, g , we obtain which completes the proof.
Journal of Applied Mathematics 5 Remark 2.5. Lemma 2.2 stays true in the discrete case.
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 cf. 10, 11 . Then we have the following Proof. The proof is a direct consequence of the discrete maximum principle.
Proof. The proof is similar to that of the continuous case. Indeed, as the basis functions of the space V h are positive, it suffices to use the discrete maximum principle.

Schwarz Alternating Methods for Nonlinear PDEs
Consider the nonlinear PDE where f · is a nondecreasing nonlinearity. Thanks to 12 , problem 2.19 has a unique solution.
Let us also assume that f · is a Lipschitz continuous on R; that is, where β is the constant defined in 2.6 . 6

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We decompose Ω into two overlapping smooth subdomains Ω 1 and Ω 2 such that 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 2.20 the following system: find 2.26

The Continuous Schwarz Sequences
Let u 0 be an initialization in C 0 Ω i.e., continuous functions vanishing on ∂Ω such that Starting from u 0 2 u 0 /Ω 2 , we respectively define the alternating Schwarz sequences u n 1 1 on Ω 1 such that u n 1 1 ∈ V u n 2 solves precisely, there exist two constants k 1 , k 2 ∈ 0; 1 which depend on Ω 1 , Γ 2 and Ω 2 , Γ 1 , respectively, such that for all n ≥ 0; 2.30

The Discretization
For i 1, 2, let τ h i be a standard regular and quasiuniform finite element triangulation in Ω i ; h i , being the meshsize. The two meshes being mutually independent Ω 1 ∩ Ω 2 , a triangle belonging to one triangulation does not necessarily belong to the other. We consider the following discrete spaces: where π h i denote an interpolation operator on Γ i .
The Discrete Maximum Principle (see [10,11]) We assume that the respective matrices resulting from the discretizations of problems 2.28 and 2.29 are M-matrices. Note that as the two meshes h 1 and h 2 are independent over the overlapping subdomains, it is impossible to formulate a global approximate problem which would be the direct discrete counterpart of problem 2.20 .

The Discrete Schwarz Sequences
Now, we define the discrete counterparts of the continuous Schwarz sequences defined in 2.28 and 2.29 .
Indeed, let u 0h be the discrete analog of u 0 , defined in 2.27 ; we, respectively, define

L ∞ -Error Analysis
This section is devoted to the proof of the main result of the present paper. To that end we begin by introducing two discrete auxiliary sequences and prove a fundamental lemma.

Two Auxiliary Schwarz Sequences
For w 0 2h u 0 2h , we define the sequences w n 1 1h and w n 1 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 2.28 , 2.29 , respectively. Then, as f · is continuous, f u n i ∞ ≤ C independent of n , and, therefore, making use of standard maximum norm estimates for linear elliptic problems, we have where C is a constant independent of both h and n.
Notation 3.1. From now on, we shall adopt the following notations: 3.4

The Main Results
The following lemma will play a key role in proving the main result of this paper.
Proof. We know from standard L ∞ -error estimate for linear problem see 13 that there exists a constant C independent of h such that Now, since 1/2 < ρ < 1, then 1 < ρ/ 1 − ρ , and therefore Let us now prove 3.5 by induction. Indeed for n 1, using the Proposition 2.7, we have in domain 1

3.8
We then have to distinguish between two cases:

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Then

3.12
Case 2 implies 3.14 So, by multiplying 3.13 by ρ we get

3.17
That is, It follows that only the case a is true, that is, 3.20 Thus

3.21
So, in both cases 1 and 2 , we have Similarly, we have in domain 2

3.26
So

3.27
while case 2 implies

3.29
So, by multiplying 3.28 by ρ we get Hence, a and b are true because they both coincide with 3.22 . So, there is either a contradiction and thus case 2 is impossible or case 2 is possible only if that is, Thus

3.37
That is, both cases 1 and 2 imply Now, let us assume that , u n 2 − u n 2h 2 .

3.41
We have again to distinguish between two cases So, a and b are true because they both coincide with 3.39 . This means that there is either contradiction and then case 2 is impossible, or case 2 is possible and then we must have ρCh 2 log h ρ u n 2 − u n Proof. Let us give the proof for i 1. The one for i 2 is similar and so will be omitted. Indeed, Let k k 1 k 2 . Then making use of Theorem 2.8 and Lemma 3.2, we get

Conclusion
We have established an error estimate for the finite element Schwarz alternating method for a nonlinear elliptic PDE on two subdomains with nonmatching grids combining a geometrical convergence result due to Lions and a standard finite element L ∞ -error analysis for linear elliptic equations.The same approach may be extended to other types of problems such as linear parabolic PDEs see 2 and singularly perturbed advection-diffusion equations see 14 .