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Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects

by X. Claeys and R. Hiptmair and E. Spindler

(Report number 2015-19)

Abstract
We consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts. Some of those may be impenetrable, giving rise to Dirichlet boundary conditions on their surfaces. We start from the second-kind boundary integral approach of $\big[$X. Claeys, and R. Hiptmair, and E. Spindler. A second-kind Galerkin boundary element method for scattering at composite objects. BIT Numerical Mathematics, 55(1):33-57, 2015$\big]$ for pure transmission problems and extend it to settings with essential boundary conditions. Based on so-called global multi-potentials, we derive variational second-kind boundary integral equations posed in $L^2(\Sigma)$, where $\Sigma$ denotes the union of material interfaces. To suppress spurious resonances, we introduce a combined-field version (CFIE) of our new method. Thorough numerical tests highlight the low and mesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces. They also confirm competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

Keywords: acoustic scattering, second-kind boundary integral equations, Galerkin boundary element methods

@Techreport{CHS15_609,
  author = {X. Claeys and R. Hiptmair and E. Spindler},
  title = {Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-19.pdf },
  year = {2015}
}

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