Report Number: CS-TR-64-6
Institution: Stanford University, Department of Computer Science
Title: A fast direct solution of Poisson's equation using Fourier analysis
Author: Hockney, Roger W.
Date: April 1964
Abstract: The demand for rapid procedures to solve Poisson's equation has lead to the development of a direct method of solution involving Fourier analysis which can solve Poisson's equation in a square region covered by a 48 x 48 mesh in 0.9 seconds on the IBM 7090. This compares favorably with the best iterative methods which would require about 10 seconds to solve the same problem. The method is applicable to rectangular regions with simple boundary conditions and the maximum observed error in the potential for several random charge distributions is $5 \times\ 10^{-7}$ of the maximum potential charge in the region.
http://i.stanford.edu/pub/cstr/reports/cs/tr/64/6/CS-TR-64-6.pdf