On the approximation of dispersive electromagnetic eigenvalue problems in 2D

TL;DR

This paper extends finite element approximation convergence results for 2D dispersive electromagnetic eigenvalue problems with sign-changing material properties, incorporating new assumptions and confirming findings through computational experiments.

Contribution

It generalizes previous meshing rules to 2D vectorial electromagnetic eigenvalue problems with dispersive materials, including both permittivity and permeability contrasts.

Findings

Theoretical convergence results are established for the eigenvalue problems.

Computational studies confirm the validity of the theoretical assumptions.

New assumptions on material contrasts are necessary for the analysis.

Abstract

We consider time-harmonic electromagnetic wave equations in composites of a dispersive material surrounded by a classical material. In certain frequency ranges this leads to sign-changing permittivity and/or permeability. Previously meshing rules were reported, which guarantee the convergence of finite element approximations to the related scalar source problems. Here we generalize these results to the electromagnetic two dimensional vectorial equations and the related holomorphic eigenvalue problems. Different than for the analysis on the continuous level, we require an assumption on both contrasts of the permittivity and the permeability. We confirm our theoretical results with computational studies.