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Geometric discretization of the multidimensional Dirac delta distribution – application to the Poisson equation with singular source terms. (English) Zbl 1380.65349

Summary: We present a discretization method for the multidimensional Dirac distribution. We show its applicability in the context of integration problems, and for discretizing Dirac-distributed source terms in Poisson equations with constant or variable diffusion coefficients. The discretization is cell-based and can thus be applied in a straightforward fashion to quadtree/octree grids. The method produces second-order accurate results for integration. Superlinear convergence is observed when it is used to model Dirac-distributed source terms in Poisson equations: the observed order of convergence is 2 or slightly smaller. The method is consistent with the discretization of Dirac delta distribution for codimension one surfaces presented in [C. Min and the second author, ibid. 226, No. 2, 1432–1443 (2007; Zbl 1125.65021); ibid. 227, No. 22, 9686–9695 (2008; Zbl 1153.65014)]. We present quadtree/octree construction procedures to preserve convergence and present various numerical examples, including multi-scale problems that are intractable with uniform grids.

MSC:

65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35A08 Fundamental solutions to PDEs
65D30 Numerical integration
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