A Local Relaxation Method for Solving Elliptic PDEs on Mesh-Connected Arrays

Abstract

A local relaxation method for solving linear elliptic PDEs with $O(N)$ processors and $O(\sqrt N )$ computation time is proposed. We first examine the implementation of traditional relaxation algorithms for solving elliptic PDEs on mesh-connected processor arrays, which require $O(N)$ processors and $O(N)$ computation time. The disadvantage of these implementations is that the determination of the acceleration factors requires some global communication at each iteration. The high communication cost increases the computation time per iteration significantly. Therefore, a local relaxation scheme is proposed to achieve the acceleration effect with very little global communication in the loading stage. We use a Fourier analysis approach to analyze the local relaxation method and also show its convergence. The convergence rate of the local relaxation method is studied by computer simulation.