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

Martin Halla, Thorsten Hohage, Florian Oberender

TL;DR

This work tackles time-harmonic scalar transmission problems with sign-changing coefficients arising from dispersive materials. It introduces a finite element discretization based on a $T$-coercive, weakly coercive reformulation obtained via a global reflection operator, enabling stable discretization with standard FE spaces for both source and dispersive eigenvalue problems. The authors provide explicit constructions and bounds for the reflection operators, along with a carefully designed quadrature strategy to handle the nonlocal transformation, and validate the approach through 2D and 3D numerical experiments, including a dispersive eigenvalue problem solved by a contour integral method. The method offers a tractable, convergent alternative to optimization-based approaches, with practical relevance for simulating surface plasmons and other sign-changing coefficient phenomena in heterogeneous, dispersive media.

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.

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

TL;DR

This work tackles time-harmonic scalar transmission problems with sign-changing coefficients arising from dispersive materials. It introduces a finite element discretization based on a -coercive, weakly coercive reformulation obtained via a global reflection operator, enabling stable discretization with standard FE spaces for both source and dispersive eigenvalue problems. The authors provide explicit constructions and bounds for the reflection operators, along with a carefully designed quadrature strategy to handle the nonlocal transformation, and validate the approach through 2D and 3D numerical experiments, including a dispersive eigenvalue problem solved by a contour integral method. The method offers a tractable, convergent alternative to optimization-based approaches, with practical relevance for simulating surface plasmons and other sign-changing coefficient phenomena in heterogeneous, dispersive media.

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 -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.
Paper Structure (12 sections, 8 theorems, 73 equations, 6 figures)

This paper contains 12 sections, 8 theorems, 73 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The weakly coercive reformulation
  3. Global reflection operators
  4. Implementation
  5. Numerical experiments
  6. Examples in two dimensions
  7. Example in three dimensions
  8. Example of a dispersive eigenvalue problem
  9. Appendix
  10. Bounds in two dimensions
  11. Bounds in three dimensions
  12. Bounds for special geometries

Key Result

Lemma 2.2

For $T_{\pm}$ be defined as above the following implication hold true:

Figures (6)

  • Figure 1: Examples for subdivided element with quadrature points
  • Figure 2: Convergence for a disc shaped domain.
  • Figure 3: Convergence for smooth triangle shaped domain.
  • Figure 4: Convergence for a ball shaped domain.
  • Figure 5: Pill shaped inclusion.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 2.1: (weak) coercivity
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4: Conditions for weak coercivity
  • proof
  • Remark 2.5
  • Remark 2.6
  • Lemma A.1
  • proof
  • ...and 7 more