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Hierarchical grid coarsening for the solution of the Poisson equation in free space. (English) Zbl 1175.35035
Summary: In many applications the solution of PDEs in infinite domains with vanishing boundary conditions at infinity is of interest. If the Green’s function of the particular PDE is known, the solution can easily be obtained by folding it with the right hand side in a finite subvolume. Unfortunately this requires $${\mathcal O}(N^2)$$ operations. T. Washio and C. W. Oosterlee [Numer. Math. 86, No. 3, 539–563 (2000; Zbl 0963.65108)]. They use infinitely many grid levels for the error analysis. In this paper we present an extension of their work. Instead of continuing the refinement process up to infinitely many grid levels, we stop the refinement process at an arbitrary level and impose the Dirichlet boundary conditions of the original problem there. The error analysis shows that the proposed method still is of order $$h^2$$, as the original method with infinitely many refinements.

##### MSC:
 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35R35 Free boundary problems for PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs
Zbl 0963.65108
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