The Geometric Solution of Laplace's Equation

Abstract

A new numerical method for the rapid solution of Laplace's equation in exterior domains and in interior domains with complicated boundaries is presented. The method is based on a formula first stated by J. J. Thomson and later refined by the authors. The mathematical foundations presented allow for the solution of field problems by means of geometric construction principles. Specifically, the method utilizes the concept of representing equipotential surfaces by polynomials for the rapid tracing of these surfaces; and is, therefore, fundamentally different from previously known techniques which are based on discretizing the domain or the boundary of the problem. For the class of problems characterized by irregular domains, the fastest available techniques have traditionally required anO(M·N) computations, whereMis the number of points inside the domain at which the solution is computed andNis the number of points used on the boundary. The new method requires anO(M) computations only and is, therefore, more advantageous in large scale calculations. This paper presents only the two-dimensional version of the geometric solution of Laplace's equation.