A convergence rate theorem for finite difference approximations to delta functions
A new rate of convergence for approximations to certain integrals over codimension one manifolds in \(\mathbb R^n\), is proved. The type of manifold is defined by the zero level set of a smooth mapping \(u:\mathbb R^n\to\mathbb R\). The approximation method used, is based on two finite difference methods for the discretizing of the delta function, originally presented by the author in a previous research work [J. Comput. Phys. 220, No. 2, 915--931 (2007; Zbl 1115.65028)], were empirical convergence rates had indicated the first and second order accuracy. In this work these convergence rates are proved for the two proposed algorithms, under fairly general hypotheses.
scientific article; zbMATH DE number 5303107
scientific article