A new numerical method for scalar eigenvalue problems in heterogeneous, dispersive, sign-changing materials

TL;DR

This paper introduces a novel finite element method for scalar eigenvalue problems in heterogeneous, dispersive, sign-changing materials, addressing challenges posed by loss of coercivity and nonlinearity in simulations.

Contribution

The paper presents a new weakly coercive reformulation and finite element scheme capable of handling complex interfaces and nonlinear eigenvalue problems in dispersive materials.

Findings

Method demonstrates stability in 2D and 3D simulations.

Convergence analysis is straightforward under ideal quadrature.

Computational experiments confirm effectiveness and accuracy.

Abstract

We consider time-harmonic scalar transmission problems between dielectric and dispersive materials with generalized Lorentz frequency laws. For certain frequency ranges such equations involve a sign-change in their principle part. Due to the resulting loss of coercivity properties, the numerical simulation of such problems is demanding. Furthermore, the related eigenvalue problems are nonlinear and give rise to additional challenges. We present a new finite element method for both of these types of problems, which is based on a weakly coercive reformulation of the PDE. The new scheme can handle $C^{1,1}$-interfaces consisting piecewise of elementary geometries. Neglecting quadrature errors, the method allows for a straightforward convergence analysis. In our implementation we apply a simple, but nonstandard quadrature rule to achieve negligible quadrature errors. We present computational experiments in 2D and 3D for both source and eigenvalue problems which confirm the stability and convergence of the new scheme.