Using domain decomposition to solve symmetric,positive-definite systems on the hypercube computer

A distributed method of solving sparse, positive-definite systems of equations, like those arising from many finite-element problems, on a hypercube computer is studied. A domain-decomposition method is introduced wherein the domain of the problem to be solved is physically split into several subdomains. Each of these subdomains is then distributed to a separate processor on the hypercube where the fill, factorization and solution of the system of equations proceeds. This physical split is based on a nodal ordering known as one-way dissection.⁴ 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 algorithm is developed to take full advantage of the inherent sparsity in the system of equations by using an envelope storage scheme. The method is applied to several geometries, and results as well as performance data for the algorithm will be given.