On the numerical treatment of singularities in solutions of laplace’s equation

Solutions to Laplace’s equation with a mixture of Dirichlet and Neumann boundary conditions can involve mild, “square-root-type” singularities at boundary points where the nature of the boundary conditions changes. An efficient finite difference method is proposed for producing accurate numerical solutions of such problems on uniform grids. The present technique is reminiscent of exponential fitting in the numerical solution of stiff ordinary differential equations. It is based on a nine-point, second-order accurate discretization of the Laplacian depending on a parameter. This discretization is fitted to the singular part of the solution by an appropriate choice of the parameter. The condition for fitting only depends on the nature of the singularity but not on the unknown magnitude of the singular component of the solution. The present approach ideally works for problems with one singularity but can easily be extended to problems with multiple singularities of the same type. The general idea is applicable to other types of singularities than the one considered in this paper. © 1995, Taylor & Francis Group, LLC. All rights reserved.