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 $\mathrm{O}(\mathrm{N}\log \mathrm{N})$ where $\mathrm{N}$ is the total number of grid points in the rectangular domain.

15 pages, 3 Figures, 1 Table