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

TL;DR

This paper demonstrates that the inverse of FEM matrices for time-harmonic Maxwell equations can be efficiently approximated using ${ m extbf{H}}$-matrices, achieving rapid convergence under standard admissibility conditions.

Contribution

It provides a theoretical proof of root exponential convergence for ${ m extbf{H}}$-matrix approximations of Maxwell FEM matrix inverses, under standard block structure criteria.

Findings

Root exponential convergence in block rank for ${ m extbf{H}}$-matrix approximations.

Effective ${ m extbf{H}}$-matrix approximation of Maxwell FEM inverses.

Theoretical validation of low-rank approximation efficiency.

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 ${\mathcal H}$-matrices. We prove that root exponential convergence in the block rank can be achieved if the block structure conforms to a standard admissibility criterion.