Finite Element Convergence Analysis of a Schwarz Alternating Method for Nonlinear Elliptic PDEs

In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for nonlinear elliptic partial differential equations in the context of linear subdomain problems and nonmatching grids. The method stands on the combination of the convergence of linear Schwarz sequences with standard finite element L  -error estimate for linear problems.


Introduction
he Schwarz alternating method can be used to solve elliptic boundary value problems on domains that consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions that results from solving a sequence of elliptic boundary value problems in each of the subdomains.
There has been extensive analysis of the Schwarz alternating method for nonlinear elliptic boundary value problems [1][2][3][4] and the references therein). Also, the effectiveness of Schwarz methods for these problems (especially those in fluid mechanics) has been demonstrated by many authors.
In this paper, we are concerned with the finite element convergence analysis of overlapping Schwarz alternating methods in the context of nonmatching grids for nonlinear PDEs, where the Schwarz sub problems are linear. This study constitutes, to some extent, an improvement of the one achieved in [5], on a Schwarz method with nonlinear sub problems.
For that, we develop an approach which combines the convergence result of Lui [6], with standard finite element error estimate for linear elliptic equations.
The rest of the paper is organized as follows. In section 2, we state the continuous alternating Schwarz sub problems and define their respective finite element counterparts in the context of nonmatching overlapping grids. In section 3, we give   L   -convergence analysis of the method.

T 2. 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 R²or R³ with sufficiently smooth boundary ∂Ω. We consider the bilinear form the right hand side: f is a regular function, where g is a regular function defined on  .
We consider the linear elliptic equation: Find The discrete counterpart of (.,.) consists of finding and h  is the Lagrange interpolation operator on ∂Ω.
On the other hand, we have 0 on Proof. The proof is a direct consequence of the discrete maximum principle.
The proof is similar to that of the continuous case. Indeed, as the basis functions s positive, it suffices to make use of the discrete maximum principle.  (11) 112 or in its weak form: Find

The Linear Schwarz Sub problems
We decompose Ω into two overlapping smooth subdomains 1  and 2  such that: u be an initial guess. We define the alternating Schwarz  (14) and the sequence   Note that Schwarz subdomain problems (14) and (15) are linear. Theorem 3. [6] The sequences (14) and (15) converge uniformly in , where u is the solution of (11).

The variational Linear Schwarz Sub problems
The corresponding variational problems read as follows: , ,

The Discretization
and let i h  denote the Lagrange interpolation operator on i  The discrete Maximum principle (see [15,16]). We assume that the respective matrices resulting from the discretization of problems (16) and (17) are M-matrices.

L  -Convergence 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 Schwarz sequences and prove a fundamental lemma.

Auxiliary Discrete Schwarz Sub problems
We construct a sequence   Then, it is clear that , .

The Main Result
The following lemma will play a key role in proving the main result of this paper.   1  1  1  1  0  0  0  0  2  2  1  1  1  1  2  2  2 1 Thus, in both cases, we have For n = 2