Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems

Abstract

In this note we present numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems in planar domains. There has recently been much progress in the development of such algorithms for linear elliptic problems ([1],[2],[3],[4]). These have focused on variational characterizations of the problem and on the preconditioning of the Schur complement associated with the decomposition. Although these could be used as part of a global Newton-type iterative scheme to solve a nonlinear problem, we choose the alternate path of first decomposing the problem and then applying an iterative method. Our motivation for this is two-fold; first, we expect it will lead to algorithms which will require less communication between subproblems, an attractive property for implementations on parallel processors; second, this approach has even been found to be more efficient for serial computations in some cases [6]. The essential step in this method is the solution of what we call the basic equations, a nonlinear analogue of the Schur complement problem. We are particularly concerned with the choice of boundary conditions at boundaries where subdomains intersect and their effect on the basic equations. Note that the methods we propose may also be applied to linear problems and seem to be novel in that context. We restrict ourselves here to the study of a decomposition into two domains: