FEM-BEM coupling for the high-frequency Helmholtz problem
Jens Markus Melenk, Ilaria Perugia, Alexander Rieder
TL;DR
The article develops a $k$-explicit analysis of FEM-BEM coupling for the high-frequency Helmholtz problem using a three-field formulation, proving quasi-optimality under scale-resolution conditions $kh/p$ small and $ rac{ ext{log}\,k}{p}$ bounded for both conforming hp-FEM and hp-DGFEM. A core innovation is a regularity-by-decomposition framework that splits adjoint solutions into a finite-regularity part with controlled $k$-growth and an analytic remainder, combined with filter operators that separate high- and low-frequency components. The analysis shows that the lower-order perturbations arising from boundary integral operators can be split into a smoothing part and an analytic part, enabling $k$-dependent stability and adjoint approximation estimates that drive the convergence results. Consequently, the paper establishes explicit, wavenumber-dependent bounds and scale-resolution criteria that guarantee quasi-optimal convergence of both discretization strategies, with potential for improved dispersion control in practical high-frequency simulations. The results unify and extend prior conforming and DG analyses, highlighting the critical role of analyticity and adjoint regularity in achieving robust, high-frequency FEM-BEM coupling.
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.