Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: A Two-Level Substructuring Preconditioner
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Date
2023-11Type
- Report
ETH Bibliography
yes
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Abstract
We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a \(2\)-dimensional triangulated surface \(\Gamma\) in \(\mathbb{R}^3\). We allow \(\Gamma\) to belong to a large class of geometries that we call polygonal multiscreens, which can be non-manifold. After introducing a new, simple conforming Galerkin discretization, we analyze a substructuring domain-decomposition preconditioner based on ideas originally developed for the Finite Element Method. The surface \(\Gamma\) is subdivided into non-overlapping regions, and the application of the preconditioner is obtained via the solution of the hypersingular equation on each patch, plus a coarse subspace correction. We prove that the condition number of the preconditioned linear system grows poly-logarithmically with \(H/h\), the ratio of the coarse mesh and fine mesh size, and our numerical results indicate that this bound is sharp. This domain-decomposition algorithm therefore guarantees significant speedups for iterative solvers, even when a large number of subdomains is used. Show more
Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichOrganisational unit
03632 - Hiptmair, Ralf / Hiptmair, Ralf
Funding
184848 - Novel Boundary Element Methods for Electromagnetics (SNF)
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ETH Bibliography
yes
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