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

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-Nedelec 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.