High Order Accurate Solution of Poisson's Equation in Infinite Domains for Smooth Functions

Abstract

In this paper a method is presented for evaluating the convolution of the Green's function for the Laplace operator with a specified function $\rho(\vec x)$ at all grid points in a rectangular domain $\Omega \subset {\mathrm R}^{d}$ ($d = 1,2,3$), i.e. a solution of Poisson's equation in an infinite domain. 4th and 6th order versions of the method achieve high accuracy when $\rho ( \vec x )$ possesses sufficiently many continuous derivatives. The method utilizes FFT's for computational efficiency and has a computational cost that is $\rm O (N \log N)$ where $\rm N$ is the total number of grid points in the rectangular domain.