FEM-BEM coupling for the high-frequency Helmholtz problem

TL;DR

This paper provides a detailed wavenumber-explicit analysis of FEM-BEM coupling methods for high-frequency Helmholtz problems, establishing conditions for quasi-optimality in conforming and DG discretizations.

Contribution

It offers a new wavenumber-explicit analysis framework for FEM-BEM coupling methods, clarifying conditions for their quasi-optimal performance at high frequencies.

Findings

Conditions $kh/p$ small and $ ext{log}(k)/p$ bounded ensure quasi-optimality.

Analysis relies on a $k$-explicit regularity theory for a three-field formulation.

Results apply to both conforming and DG discretizations.

Abstract

We present a wavenumber-explicit analysis of FEM-BEM coupling methods for time-harmonic Helmholtz problems proposed in arXiv:2004.03523 for conforming discretizations and in arXiv:2105.06173 for discontinuous Galerkin (DG) volume discretizations. We show that the conditions that $kh/p$ be sufficiently small and that $\log(k) / p$ be bounded imply quasi-optimality of both conforming and DG-method, where $k$ is the wavenumber, $h$ the mesh size, and $p$ the approximation order. The analysis relies on a $k$-explicit regularity theory for a three-field coupling formulation.