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On the evaluation of layer potentials close to their sources

Helsing, Johan LU and Ojala, Rikard LU (2008) In Journal of Computational Physics 227(5). p.2899-2921
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Computational Physics
volume
227
issue
5
pages
2899 - 2921
publisher
Elsevier
external identifiers
  • wos:000253598700009
  • scopus:38549083654
ISSN
0021-9991
DOI
10.1016/j.jcp.2007.11.024
language
English
LU publication?
yes
id
e0869bd4-bc9e-41c5-b40c-834e6ab9c6b5 (old id 629781)
alternative location
http://www.maths.lth.se/na/staff/helsing/JCP08a.pdf
date added to LUP
2008-01-17 16:46:04
date last changed
2017-05-28 04:40:11
@article{e0869bd4-bc9e-41c5-b40c-834e6ab9c6b5,
  abstract     = {When solving elliptic boundary value problems using integral<br/><br>
 equation methods one may need to evaluate potentials represented by<br/><br>
 a convolution of discretized layer density sources against a kernel.<br/><br>
 Standard quadrature accelerated with a fast hierarchical method for<br/><br>
 potential field evaluation gives accurate results far away from the<br/><br>
 sources. Close to the sources this is not so. Cancellation and<br/><br>
 nearly singular kernels may cause serious degradation. This paper<br/><br>
 presents a new scheme based on a mix of composite polynomial<br/><br>
 quadrature, layer density interpolation, kernel approximation,<br/><br>
 rational quadrature, high polynomial order corrected interpolation<br/><br>
 and differentiation, temporary panel mergers and splits, and a<br/><br>
 particular implementation of the GMRES solver. Criteria for which<br/><br>
 mix is fastest and most accurate in various situations are also<br/><br>
 supplied. The paper focuses on the solution of the Dirichlet problem<br/><br>
 for Laplace's equation in the plane. In a series of examples we<br/><br>
 demonstrate the efficiency of the new scheme for interior domains<br/><br>
 and domains exterior to up to 2000 close-to-touching contours.<br/><br>
 Densities are computed and potentials are evaluated, rapidly and<br/><br>
 accurate to almost machine precision, at points that lie arbitrarily<br/><br>
 close to the boundaries.},
  author       = {Helsing, Johan and Ojala, Rikard},
  issn         = {0021-9991},
  language     = {eng},
  number       = {5},
  pages        = {2899--2921},
  publisher    = {Elsevier},
  series       = {Journal of Computational Physics},
  title        = {On the evaluation of layer potentials close to their sources},
  url          = {http://dx.doi.org/10.1016/j.jcp.2007.11.024},
  volume       = {227},
  year         = {2008},
}