Electromagnetic Steklov eigenvalues: approximation analysis

Abstract

We continue the work of Camano et al. [SIAM J. Math. Anal. 49 (2017) 4376–4401] on electromagnetic Steklov eigenvalues. The authors recognized that in general the eigenvalues do 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 Steklov eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function A(⋅). We construct a “test function operator function” T(⋅) so that A(λ) is weakly T(λ)-coercive for all suitable λ, i.e. T(λ)*A(λ) is a compact perturbation of a coercive operator. The construction of T(⋅) 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 which concerns the tangential trace. The existence and construction of such projection operators remain open questions.