Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: A Two-Level Substructuring Preconditioner

by M. Averseng and X. Claeys and R. Hiptmair

(Report number 2023-40)

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.

Keywords: Multi-Screens, Boundary Element Method, Domain Decomposition, Substructuring Preconditioner

@Techreport{ACH23_1077,
  author = {M. Averseng and X. Claeys and R. Hiptmair},
  title = {Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: A Two-Level Substructuring Preconditioner},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-40.pdf },
  year = {2023}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).