THE RELAXATION TREATMENT OF SINGULAR POINTS IN POISSON'S EQUATION

Abstract

If Φ is harmonic or is a solution to Poisson's equation, it may have singular points in the field or on the boundary at which it (a) has finite values, but has infinite derivatives, (b) has logarithmic infinities, or (c) has simple discontinuities. This paper describes methods that can be adopted by computers using the relaxation technique when working in the neighbourhood of these points. Methods of obtaining accurate derivatives and integrals in these neighbourhoods are also given. Four examples illustrate the methods and suggest that the accuracy is comparable with that obtained in similar problems without singularities.