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Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk

by M. Averseng and F. Alouges

(Report number 2022-32)

Abstract
We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in R3. Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolutions and can thus be evaluated cheaply in the context of iterative methods. For the Laplace equation (i.e. for the wavenumber \(k=0\)) with Dirichlet condition on the disk and on regular meshes, we prove that the preconditioned linear system has a bounded condition number uniformly in the mesh size. We further provide numerical evidence indicating that the preconditioners also perform well for large values of \(k\) and on locally refined meshes.

Keywords: Boudary Element Methods, Preconditioning, Singularities in PDEs

@Techreport{AA22_1020,
  author = {M. Averseng and F. Alouges},
  title = {Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-32},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-32.pdf },
  year = {2022}
}

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