Electromagnetic Stekloff eigenvalues: approximation analysis

TL;DR

This paper analyzes the approximation of electromagnetic Stekloff eigenvalues, introducing a framework using holomorphic operator functions and T-compatible Galerkin methods to ensure convergence of eigenvalue approximations.

Contribution

It extends previous work by formulating the eigenvalue problems as holomorphic operator functions and establishing conditions for convergence of Galerkin approximations.

Findings

Constructed a test function operator function T(·) for weak T(·)-coercivity.

Proved convergence of Galerkin approximations for the modified problem.

Identified open questions regarding projection operators for the original problem.

Abstract

We continue the work of [Camano, Lackner, Monk, SIAM J. Math. Anal., Vol. 49, No. 6, pp. 4376-4401 (2017)] on electromagnetic Stekloff eigenvalues. The authors recognized that in general the eigenvalues due not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The present article considers the original and the modified electromagnetic Stekloff eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function $A(\cdot)$. We construct a "test function operator function" $T(\cdot)$ so that $A(\lambda)$ is weakly $T(\lambda)$-coercive for all suitable $\lambda$, i.e. $T(\lambda)^*A(\lambda)$ is a compact perturbation of a coercive operator. The construction of $T(\cdot)$ relies on a suitable decomposition of the function space into subspaces and an apt sign change on each subspace. For the approximation analysis, we apply the framework of T-compatible Galerkin approximations. For the modified problem, we prove that convenient commuting projection operators imply T-compatibility and hence convergence. For the original problem, we require the projection operators to satisfy an additional commutator property involving the tangential trace. The existence and construction of such projection operators remain open questions.