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Zbl 1135.65404
Helsing, Johan; Ojala, Rikard
On the evaluation of layer potentials close to their sources. (English)
[J] J. Comput. Phys. 227, No. 5, 2899-2921 (2008). ISSN 0021-9991

Summary: When solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel. Standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources. Close to the sources this is not so. Cancellation and nearly singular kernels may cause serious degradation. This paper presents a new scheme based on a mix of composite polynomial quadrature, layer density interpolation, kernel approximation, rational quadrature, high polynomial order corrected interpolation and differentiation, temporary panel mergers and splits, and a particular implementation of the GMRES solver. Criteria for which mix is fastest and most accurate in various situations are also supplied. The paper focuses on the solution of the Dirichlet problem for Laplace's equation in the plane. In a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close-to-touching contours. Densities are computed and potentials are evaluated, rapidly and accurate to almost machine precision, at points that lie arbitrarily close to the boundaries.

MSC 2000:: *65N38 Boundary element methods (BVP of PDE)
35J05 Laplace equation, etc.

Keywords: numerical examples; potential theory; fast solvers; singular integrals; multiply connected domains; Dirichlet-Neumann map; elliptic boundary value problems; integral equation methods; polynomial quadrature; layer density interpolation; kernel approximation; rational quadrature; GMRES; Dirichlet problem; Laplace's equation

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