The treatment of singularities of partial differential equations by relaxation methods

Abstract

In the course of a study of boundary value problems arising in radiation theory and electrostatics, the treatment of singularities demanded special attention. In most problems of practicar importance boundaries with sharp corners occur. Such sharp corners give rise to singularities of various types. When the com- puted function is bounded, but has a branch point at the corner, the difficulty is not serious. The use of a graded net with a finer mesh size near the corner is possible. Conformai transformation which automatically provides a finer net near corners is also successful. The mesh size near the corner should be of the order of magnitude of the radius of curvature of the corner, and when this is small a mathematical idealiza- tion involving infinitely sharp corners is preferable. The special treatment outlined in this note makes use of such an idealization and shortens the labour considerably. Special treatment is essential when the function approaches infinite values near the corner.