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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:227

Article

Helsing, Johan¹  Ojala, Rikard¹ 

[1] Lund Univ, Ctr Math Sci, SE-22100 Lund, Sweden

关键词: Corner singularity;    Multiphase material;    Triple-junction;    Integral equation;    Mesh grading;    Conductivity;   

DOI  :  10.1016/j.jcp.2008.06.022

来源: Elsevier

【 摘 要 】

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). (c) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free