Electromagnetic Stekloff eigenvalues: approximation analysis

09/02/2019
by   Martin Halla, et al.
0

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(·). 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 involving the tangential trace. The existence and construction of such projection operators remain open questions.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/01/2022

Regular Convergence and Finite Element Methods for Eigenvalue Problems

Regular convergence, together with various other types of convergence, h...
research
01/04/2022

On an eigenvalue property of Summation-By-Parts operators

Summation-By-Parts (SBP) methods provide a systematic way of constructin...
research
08/14/2019

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

We consider Galerkin approximations of holomorphic Fredholm operator eig...
research
07/19/2020

Analysis of radial complex scaling methods: scalar resonance problems

We consider radial complex scaling/perfectly matched layer methods for s...
research
09/06/2021

Broken-FEEC approximations of Hodge Laplace problems

In this article we study nonconforming discretizations of Hilbert comple...
research
10/06/2021

A structure preserving shift-invert infinite Arnoldi algorithm for a class of delay eigenvalue problems with Hamiltonian symmetry

In this work we consider a class of non-linear eigenvalue problems that ...
research
10/06/2021

Computational lower bounds of the Maxwell eigenvalues

A method to compute guaranteed lower bounds to the eigenvalues of the Ma...

Please sign up or login with your details

Forgot password? Click here to reset