Fast Parallel Iterative Solution of Poisson’s and the Biharmonic Equations on Irregular Regions

In [SIAM J. Numer. Anal., 21 (1984), pp. 285–299], a method was introduced for solving Poisson’s or the biharmonic equation on an irregular region by making use of an integral equation formulation. Because fast solvers were used to extend the solution to an enclosing rectangle, this method avoided many of the standard problems associated with integral equations. The equations that arose were Fredholm integral equations of the second kind with bounded kernels. In this paper iterative methods are used to solve the dense nonsymmetric linear systems arising from the integral equations. Because the matrices are very well conditioned, conjugate gradient-like methods can be used and will converge very rapidly. The methods are very amenable to vectorization and parallelization, and parallel and vector implementations are described on shared memory multiprocessors. Numerical experiments are described and results presented for a three-dimensional interface problem for the Laplacian on a recording head geometry.