Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

Xavier Claeys 1, 2 Ralf Hiptmair 3
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, Institut National des Sciences Mathématiques et de leurs Interactions
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, Numer Math, 100, 1–19 (2005)] 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, Commun Pure and Appl Math, 66, 1163–1201 (2013)] to situations with Dirichlet boundary conditions.
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Xavier Claeys, Ralf Hiptmair. Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects. Integral Equations and Operator Theory, Springer Verlag, 2015, 81 (2), pp.151-189. ⟨10.1007 / s00020-014-2197-y⟩. ⟨hal-01094440⟩

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