Boundary method-based domain decomposition on multiprocessors

Abstract

An efficient method has been developed for the fast solution of the boundary problems of Poisson's equation on irregular as well as regular domains. The method, called the boundary method-based domain decomposition or BMDD, combines the attractiveness of the domain decomposition technique in parallel solution of the boundary value problem with the advantage of a boundary method. Unlike the Schwarz alternating method or the iterative substructuring method, where the interface values are usually solved by Preconditioned Conjugate Gradient iteration which requires subdomain solvers for all subdomains at each step, in the BMDD approach the interface values are evaluated after an approximate solution in an explicit form is obtained by a boundary method. Our method is suitable for parallel processing, because the dominant part of computation is solving completely independent subproblems, and computation of the interface values by a boundary method also involves trivial parallelization in matrix generation and in evaluation of interface values. The HPA (harmonic polynomial approximation), including the AHPA (augmented HPA), has been identified as a preferred boundary method to be used with BMDD, based on our analysis and numerical experiments. A new parallel Poisson solver has been obtained, which consists of a Poisson kernel method-like parallel Laplace solver on an irregular domain and a parallel Poisson solver on a disk based on integral formulation.