Using Domain Decomposition to Solve Positive-Definite Systems on the Hypercube Computer

Abstract

A distributed method of solving sparse, positive-definite systems of equations on a hypercube computer, like those arising fiom many finite-element problems, is studied. A domain decomposition method is introduced wherein the domain of the problem to be solved is physically split into several sub-domains. This physical split is based on an ordering known as one-way dissection [ I ] . The one-way dissection ordering generates a block-diagonal system of equations which is well suited to a parallel implementation. Once the ordering has been accomplished each of the subdomains is then distributed to a processor in the hypercube computer as necessary. The method is applied to two-dimensional electrostatic problems which are governed by Laplace’s equation. Since the finite-element method is used to discretize the problem the method is developed to take full advantage of the inherent sparsity. The algorithm is applied to several geometries.