A spectral boundary element method for acoustic interference problems

TL;DR

This paper introduces a spectral boundary element method for high-frequency acoustic transmission problems with complex interface behaviors, providing a reliable and efficient numerical approach with proven convergence and confirmed by numerical experiments.

Contribution

It develops a spectral Galerkin boundary element method for Helmholtz equations with jumping coefficients, including convergence analysis and numerical validation.

Findings

The integral equation is well posed.

Convergence is quasi-optimal under certain degrees of freedom.

Numerical experiments confirm efficiency and theoretical sharpness.

Abstract

In this paper we consider high-frequency acoustic transmission problems with jumping coefficients modelled by Helmholtz equations. The solution then is highly oscillatory and, in addition, may be localized in a very small vicinity of interfaces (whispering gallery modes). For the reliable numerical approximation a) the PDE is tranformed in a classical single trace integral equation on the interfaces and b) a spectral Galerkin boundary element method is employed for its solution. We show that the resulting integral equation is well posed and analyze the convergence of the boundary element method for the particular case of concentric circular interfaces. We prove a condition on the number of degrees of freedom for quasi-optimal convergence. Numerical experiments confirm the efficiency of our method and the sharpness of the theoretical estimates.