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Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning

Helsing, Johan LU and Ojala, Rikard LU (2008) In Journal of Computational Physics 227(20). p.8820-8840
Abstract
We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners... (More)
We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10-12. In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of 10-14. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Corner singularity Multiphase material Triple-junction Integral equation Mesh grading Conductivity
in
Journal of Computational Physics
volume
227
issue
20
pages
8820 - 8840
publisher
Elsevier
external identifiers
  • wos:000260267700005
  • scopus:50249139169
ISSN
0021-9991
DOI
10.1016/j.jcp.2008.06.022
language
English
LU publication?
yes
id
299477ff-86b0-48e5-92bd-3dd67cca0d37 (old id 1223893)
alternative location
http://www.maths.lth.se/na/staff/helsing/JCP08b.pdf
date added to LUP
2008-10-06 16:29:59
date last changed
2017-10-01 03:50:46
@article{299477ff-86b0-48e5-92bd-3dd67cca0d37,
  abstract     = {We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10-12. In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of 10-14.},
  author       = {Helsing, Johan and Ojala, Rikard},
  issn         = {0021-9991},
  keyword      = {Corner singularity Multiphase material Triple-junction Integral equation Mesh grading Conductivity},
  language     = {eng},
  number       = {20},
  pages        = {8820--8840},
  publisher    = {Elsevier},
  series       = {Journal of Computational Physics},
  title        = {Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning},
  url          = {http://dx.doi.org/10.1016/j.jcp.2008.06.022},
  volume       = {227},
  year         = {2008},
}