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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–Nédélec coupling. For each coupling, we provide discrete interior regularity estimates. As a consequence, we are able to prove the existence of exponentially convergent    $$\mathcal {H}$$  H -matrix approximants to the inverse matrices corresponding to the lowest order Galerkin discretizations of the couplings.

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