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

Abstract

Abstract We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle Ω − ⊂ R d \Omega^{-}\subset\mathbb{R}^{d} , d = 2 , 3 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 ∂ Ω − \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.