Caccioppoli-type estimates and $\mathcal{H}$-Matrix approximations to inverses for FEM-BEM couplings

TL;DR

This paper develops new regularity estimates for FEM-BEM couplings and demonstrates the exponential convergence of $\\mathcal{H}$-matrix approximations to their inverse matrices, enhancing computational efficiency.

Contribution

It introduces discrete interior regularity estimates for three FEM-BEM coupling methods and proves exponential convergence of $\\mathcal{H}$-matrix approximants to their inverses.

Findings

Discrete interior regularity estimates for FEM-BEM couplings

Existence of exponentially convergent $\\mathcal{H}$-matrix approximants

Improved computational efficiency for coupled FEM-BEM systems

Abstract

We consider three different methods for the coupling of the finite element method and the boundary element method, the Bielak-MacCamy coupling, the symmetric coupling, and the Johnson-N\'ed\'elec coupling. For each coupling we provide discrete interior regularity estimates. As a consequence, we are able to prove the existence of exponentially convergent $\mathcal{H}$-matrix approximants to the inverse matrices corresponding to the lowest order Galerkin discretizations of the couplings.