Some experiments with singularities in linear elliptic partial differential equations

Abstract

Error analysis for finite-difference methods reveals that very small intervals will be needed for good accuracy in the neighbourhood of boundary singularities in elliptic equations. Methods are discussed for reducing the labour by combining finite-difference equations remote from singularities with special formulae taking implicit account of the singularities. The analysis of the latter is also outlined, and two practical examples are treated, one with an isolated singularity in a problem with Cartesian coordinates and one with two singularities in a problem with cylindrical polar coordinates. Free boundary problems are considered briefly. Comparison is made with previous methods and some suggestions for further research are included.