The numerical solution of elliptic and parabolic partial differential equations with boundary singularities

Abstract

A general numerical method is described for the solution of linear elliptic and parabolic partial differential equations in the presence of boundary singularities. The method is suitable for use with either a finite-difference or finite element scheme. Modified approximations for the derivatives are developed using the local analytical form of the singularity. General guidelines are given showing how the local analytical form can be found and how the modified approximations can be developed for many problems of mathematical physics. These guidelines are based on the reduction of the differential equation to the form Δu = gu + f. The potential problem treated by Motz and Woods is taken as a numerical example. The numerical results compare favourably with those obtained by other techniques.