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Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

by I. Labarca and R. Hiptmair

(Report number 2022-24)

Abstract
We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle \(\Omega^- \subset \mathbb{R}^d, \ d=2,3\). By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann-Schwinger equation. Besides, it features integral operators defined on \( \partial\Omega^- \) and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

Keywords: volume integral equations, boundary integral equations, acoustic scattering

@Techreport{LH22_1012,
  author = {I. Labarca and R. Hiptmair},
  title = {Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-24.pdf },
  year = {2022}
}

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