On the evaluation of layer potentials close to their sources

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Abstract

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.

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Subject classification (UKÄ) – MANDATORY

  • Mathematics
Original languageEnglish
Pages (from-to)2899-2921
JournalJournal of Computational Physics
Volume227
Issue number5
Publication statusPublished - 2008
Publication categoryResearch
Peer-reviewedYes

Bibliographic note

The paper appeared electronically November 28, 2007, and subsequently in the paper issue of the journal February 20, 2008. The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)

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