$\mathcal{H}$-matrix approximability of inverses of FEM matrices for the time-harmonic Maxwell equations

Abstract

The inverse of the stiffness matrix of the time-harmonic Maxwell equation with perfectly conducting boundary conditions is approximated in the blockwise low-rank format of H-matrices. We prove that root exponential convergence in the block rank can be achieved if the block structure conforms to a standard admissibility criterion.

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.…”