Acceleration of the Schwarz Method for Elliptic Problems

Abstract

In this paper, we present a family of domain decomposition methods based on Aitken-like acceleration of the Schwarz method, which is an iterative procedure with linear rate of convergence. This paper is a generalization of our method first introduced in 2000 that was restricted to Cartesian grids. We consider the finite volume (FV) approximation on general quadrangle meshes introduced by Faille in 1992. We first present the so-called Aitken--Schwarz procedure for a linear differential operator. This solver is a direct solver, but its computational cost in the general case might be prohibitive. Second, we introduce the Steffensen--Schwarz variant, which is an iterative domain decomposition solver that can be applied to linear and nonlinear problems. We show that this iterative solver has reasonable numerical efficiency and can be applied successfully to several classes of linear and nonlinear elliptic problems. However, the salient feature of our method is that our algorithm has high tolerance for slow networks in the context of distributed parallel computing and is attractive to use with computer architectures for which performance is limited by the memory bandwidth rather than the flop performance of the CPU. This is currently the case for most parallel computers using RISC processor architecture, such as Beowulf systems.