The Aitken-Like Acceleration of the Schwarz Method on Nonuniform Cartesian Grids

In this paper, we present a family of domain decomposition based on an Aitken like acceleration of the Schwarz method seen as an iterative procedure with linear rate of convergence. This paper is a generalization of the method first introduced at the 12 International Conference on Domain Decomposition that was restricted to regular Cartesian grids. The potential of this method to provide scalable parallel computing on a geographically broad grid of parallel computers was demonstrated for some linear and nonlinear elliptic problems discretized by finite differences on a Cartesian mesh. The main thrust of this paper is to present a generalization of the method to non-uniform Cartesian meshes. The salient feature of the method consists of accelerating the sequence of traces on the artificial interfaces generated by the Schwarz procedure using a good approximation of the main eigenvectors of the trace transfer operator. For linear separable elliptic operators, our solver is a direct solver. For nonlinear operators, we use an approximation of the eigenvectors of the Jacobian of the trace transfer operator. The acceleration is then applied to the sequence generated by the Schwarz algorithm applied directly to the nonlinear operator.