A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

Abstract

In this paper we propose a direct method for the solution of the Poisson equation in rectangular regions. It has an arbitrary order accuracy and low CPU requirements which makes it practical for treating large-scale problems. The method is based on a pseudospectral Fourier approximation and a polynomial subtraction technique. Fast convergence of the Fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions. These functions have the same discontinuities at the corner points as the sought solution. In addition to this, they satisfy the Laplace equation so that the subtraction procedure does not generate nonperiodic, nonhomogeneous terms. The solution of a boundary value problem is obtained in a series form in O(N log N) floating point operations, where N2 is the number of grid nodes. Evaluating the solution at all N2 interior points requires O(N2 log N) operations.