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

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

Exponential meshes and $\mathcal{H}$-matrices

“…This approach generalizes to certain classes of pseudodifferential operators, [DHS17], and results in exponential convergence in the block rank up the final projection error. A fully discrete approach, which avoids the final projection steps and leads to exponential convergence in the block rank, was taken in [FMP15,FMP21] in a FEM setting on quasi-uniform meshes and in the boundary element method (BEM) in [FMP16,FMP17,FMP20]. The generalization of [FMP15] to non-uniform meshes was achieved in [AFM21a] for low order FEM on certain classes of meshes that includes algebraically graded meshes.…”

$\mathcal{H}$-inverses for RBF interpolation

“…Yet, a basic question remains whether the target, i.e., the inverse of the system matrix or the LU-factorization, can be represented in the H-matrix format. This question has been answered for finite element discretizations of elliptic PDEs in [BH03], recently improved in [FMP15] (to arbitrary accuracy) and [AFM21] (locally refined meshes), as well as for boundary element discretizations, [FMP16,FMP17], and the coupling of finite elements and boundary elements, [FMP20].…”