Regularization Techniques for Singular Source Terms in Differential Equations

Abstract

Regularization of singular source terms in partial differential equations is a widely used approach to deal with the challenge of numerically approximating such equations. In this paper, we analyze regularization techniques in one and multi-dimensions. We consider general numerical grids as well as the special case of a uniform grid, in which case discrete properties allow for reduced errors.

All regularizations are based on a regularization of the Dirac delta function in one dimension, that are extended to more dimensions. It is critical how this extension is done. While it can produce a multi-dimensional approximation with the same accuracy as given by the one-dimensional analysis, we also show that a technique commonly used in connection to level-set methods may lead to O(1) errors. Modifications to this inconsistent technique are introduced.