A convergence rate theorem for finite difference approximations to delta functions

We prove a rate of convergence theorem for approximations to certain integrals over codimension one manifolds in RnRn. The type of manifold involved here is defined by the zero level set of a smooth mapping u:Rn↦Ru:Rn↦R. Our approximations are based on the two finite difference methods for discretizing delta functions presented in [16]. We included a convergence proof in that paper, but only proved rates of convergence in some greatly simplified situations. Numerical experiments indicated that our two methods were at least first and second order accurate, respectively. In this note we prove those empirical convergence rates for the two algorithms under fairly general hypotheses.