Two methods for discretizing a delta function supported on a level set (Q868251)

Let \(f:{\mathbb R}^n \to {\mathbb R}\) and \(u:{\mathbb R}^n \to {\mathbb R}\) be smooth functions which are given by their data on a grid. Let \(\Gamma\) be the zero level set of \(u\). The author considers the problem of approximating the integral \(\int_{\Gamma} f(x)\,ds\). It is common practice to replace the integral above by \[ \int_{{\mathbb R}^n} f(x)\, \delta(u(x))\,\| \nabla u(x)\| \, dx, \] where \(\delta\) denotes the Dirac delta function. Then one approximates this integral using the available grid-defined function values. The author proposes two methods for discretization of \(\delta(u(x))\).