Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning
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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 nonstandard 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 1012. In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of 1014.
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Original language  English 

Pages (fromto)  88208840 
Journal  Journal of Computational Physics 
Volume  227 
Issue number  20 
Publication status  Published  2008 
Publication category  Research 
Peerreviewed  Yes 
Bibliographic note
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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