The Numerical Solution of Laplace’s Equation in Three Dimensions

We consider the Dirichlet problem for Laplace’s equation, on a simply-connected three-dimensional region with a smooth boundary. This problem is easily converted to the solution of a Fredholm integral equation of the second kind, based on representing the harmonic solution as a double layer potential function. We solve this integral equation formulation by using Galerkin’s method, with spherical harmonics as the basis functions. This approach leads to small linear systems, and once the Galerkin coefficients for the region have been calculated, the computation time is small for any particular boundary function. The major disadvantage of the method is the calculation of the Galerkin coefficients, each of which is a four-fold integral with a singular integrand. Theoretical and computational details of the method are presented.