On the approximation of dispersive electromagnetic eigenvalue problems in two dimensions.

The article focuses on the analysis and finite element approximation of time-harmonic electromagnetic wave equations in two-dimensional composites consisting of dispersive and classical materials, where the permittivity and permeability may change sign depending on frequency. It extends previous scalar results to vectorial Maxwell equations and associated holomorphic eigenvalue problems, introducing a novel construction of a bijective operator \(T(\lambda)\) based on local reflection operators for each Helmholtz component to establish weak \(T\)-coercivity and Fredholm properties of the operator \(A(\lambda)\). The study proves that h-finite element methods with Nédélec elements on locally \(R\)-conform meshes are \(T\)-compatible, guaranteeing convergence of both source and eigenvalue problem approximations under assumptions on the contrasts of permittivity and permeability relative to a critical interval determined by interface corner angles. Computational experiments confirm the theoretical findings and indicate that the finite element method performs reliably even beyond the theoretical contrast assumptions.