Two methods for discretizing a delta function supported on a level set

This paper presents two new methods for discretizing a Dirac delta function which is concentrated on the zero level set of a smooth function u: Rn ↦ R. The function u is only known at the discrete set of points belonging to a regular mesh covering Rn. These two methods are used to approximate integrals over the manifold defined by the level set. Both methods are conceptually simple and easy to implement. We present the results of numerical experiments indicating that as the mesh size h goes to zero, the rate of convergence is at least O(h) for the first method, and O(h2) for the second method. We perform a limited analysis of the proposed algorithms, including a proof of convergence for both methods.