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Two methods for discretizing a delta function supported on a level set. (English) Zbl 1115.65028
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 $$\delta(u(x))$$.

##### MSC:
 65D32 Numerical quadrature and cubature formulas 58C35 Integration on manifolds; measures on manifolds 46F10 Operations with distributions and generalized functions 41A55 Approximate quadratures
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##### References:
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