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Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

by X. Claeys and R. Hiptmair

(Report number 2014-06)

Abstract
We study direct first-kind boundary integral equations arising from transmission problems for the Helmholtz equation with piecewise constant coefficients and Dirichlet boundary conditions imposed on a closed surface. We identify necessary and sufficient conditions for the occurrence of so-called spurious resonances, that is, the failure of the boundary integral equations to possess unique solutions. Following \([\)A. Buffa and R. Hiptmair, Regularized combined field integral equations, Numer. Math., 100 (2005), pp. 1-19\(]\) we propose a modified version of the boundary integral equations that is immune to spurious resonances. Via a gap construction it will serve as the basis for a universally well-posed stabilized global multi-trace formulation that generalizes the method of \([\) X. Claeys and R. Hiptmair, Multi-trace boundary integral formulation for acoustic scattering by composite structures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 1163-1201\(]\) to situations with Dirichlet boundary conditions.

Keywords: Acoustic scattering, Helmholtz equation, boundary integral equations (BIE), single-trace BIE, combined field integral equations (CFIE), global multi-trace BIE

@Techreport{CH14_556,
  author = {X. Claeys and R. Hiptmair},
  title = {Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-06.pdf },
  year = {2014}
}

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