On some Aitken‐like acceleration of the Schwarz method

Abstract

AbstractIn this paper we present a family of domain decomposition based on Aitken‐like acceleration of the Schwarz method seen as an iterative procedure with a linear rate of convergence. We first present the so‐called Aitken–Schwarz procedure for linear differential operators. The solver can be a direct solver when applied to the Helmholtz problem with five‐point finite difference scheme on regular grids. We then introduce the Steffensen–Schwarz variant which is an iterative domain decomposition solver that can be applied to linear and nonlinear problems. We show that these solvers have reasonable numerical efficiency compared to classical fast solvers for the Poisson problem or multigrids for more general linear and nonlinear elliptic problems. However, the salient feature of our method is that our algorithm has high tolerance to slow network in the context of distributed parallel computing and is attractive, generally speaking, to use with computer architecture for which performance is limited by the memory bandwidth rather than the flop performance of the CPU. This is nowadays the case for most parallel. computer using the RISC processor architecture. We will illustrate this highly desirable property of our algorithm with large‐scale computing experiments. Copyright © 2002 John Wiley & Sons, Ltd.