The numerical approximation of a delta function with application to level set methods

It is shown that a discrete delta function can be constructed using a technique developed by Anita Mayo [The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Sci. Comput. 21 (1984) 285–299] for the numerical solution of elliptic equations with discontinuous source terms. This delta function is concentrated on the zero level set of a continuous function. In two space dimensions, this corresponds to a line and a surface in three space dimensions. Delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function. The numerical implementation of these delta functions achieves the expected order of accuracy.