The Direct Solution of the Discrete Poisson Equation on a Disk

Abstract

A direct method is developed for obtaining the discrete solution of the polar coordinate form of Poisson’s equation defined on a disk. The problem is solved subject to both Dirichlet and Neumann boundary conditions. For the Dirichlet boundary condition, the solution is obtained as the superposition of two solutions defined on an annulus. The direct method may be used to obtain one and a short additional calculation provides the other. For the Neumann boundary condition a solution may not exist; however, a method is given for obtaining a solution in the sense of least squares.