Parallel Poisson and Biharmonic solvers

Abstract

In this paper we develop direct and iterative algorithms for the solution of finite difference approximations of the Poisson and Biharmonic equations on a square, using a number of arithmetic units in parallel. Assuming ann×n grid of mesh points, we show that direct algorithms for the Poisson and Biharmonic equations require 0(logn) and 0(n) time steps, respectively. The corresponding speedup over the sequential algorithms are 0(n 2) and 0(n 2logn). We also compare the efficiency of these direct algorithms with parallel SOR and ADI algorithms for the Poisson equation, and a parallel semi-direct method for the Biharmonic equation treated as a coupled pair of Poisson equations.