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Stable FEM-BEM coupling for Helmholtz transmission problems

by R. Hiptmair and P. Meury

(Report number 2005-06)

Abstract
We consider acoustic scattering at a non-smooth penetrable object and coupled boundary element finite element schemes for its numerical simulation. Straightforward coupling approaches are haunted by instabilities at wave numbers related to interior resonances, the so-called spurious resonances. A remedy is offered by adopting the idea underlying the widely used combined field integral equations. We apply it in the form of modified trace operators. These will also feature regularizing operators to offset the lack of compactness of the double-layer potential integral operators on non-smooth surfaces. Calderón projectors can be defined based on the modified trace operators. Thus, Costabel's approach to the symmetric coupling of domain variational formulations and boundary integral equations carries over. The modified traces guarantee uniqueness of solutions of the coupled problem, whereas regularization ensures coercivity. From this we immediately conclude asymptotic quasi-optimality of a combined finite element and boundary element Galerkin discretization.

Keywords:

@Techreport{HM05_345,
  author = {R. Hiptmair and P. Meury},
  title = {Stable FEM-BEM coupling for Helmholtz transmission problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2005-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2005/2005-06.pdf },
  year = {2005}
}

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