Volume 55, Number 1, January-February 2021
|Page(s)||57 - 76|
|Published online||18 February 2021|
Electromagnetic Steklov eigenvalues: approximation analysis
Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Deutschland
2 Institut für Numerik und Angewandte Mathematik, Geog-August-Universität Göttingen, Lotzestraße 16-18, 37083 Göttingen, Deutschland
* Corresponding author: email@example.com
Accepted: 10 November 2020
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.
Mathematics Subject Classification: 35J25 / 35R30 / 65H17 / 65N25
Key words: Steklov eigenvalues / nondestructive testing / T-coercivity
© EDP Sciences, SMAI 2021
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