Regularization of the Dirichlet Problem for Laplace's Equation: Surfaces of Revolution

Abstract

Based on the idea of analytical regularization, a mathematically rigorous and numerically efficient method to solve the Laplace equation with a Dirichlet boundary condition on an open or closed arbitrarily shaped surface of revolution is described. To improve the convergence of the series for the single-layer density, we extracted and evaluated in an explicit form the singularity of the density at the surface edge. Numerical investigations of canonical structures, such as the open prolate spheroid and the open surface obtained by the rotation of Pascal's Limacon or the Cassini Oval, exhibit the high accuracy and wide applicability of the method.