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Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning. (English) Zbl 1152.65114

Summary: 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 ${10}^{-14}$.

MSC:
 65N38 Boundary element methods (BVP of PDE) 35Q60 PDEs in connection with optics and electromagnetic theory 78A30 Electro- and magnetostatics 65F35 Matrix norms, conditioning, scaling (numerical linear algebra) 78M15 Boundary element methods (optics)
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