Spectral analysis and domain truncation methods for Maxwell's equations

TL;DR

This paper investigates how the spectrum of the anisotropic Maxwell system with bounded conductivity is approximated by domain truncation, providing new spectral enclosure results and criteria for eigenvalue behavior.

Contribution

It introduces a novel non-convex spectral enclosure for Maxwell's equations and develops abstract results on the essential spectrum of polynomial pencils and block operator matrices.

Findings

Established a new spectral enclosure for Maxwell's system.

Provided criteria for non-accumulation of eigenvalues on the imaginary axis.

Showed spectral pollution occurs only in the essential numerical range for asymptotically constant coefficients.

Abstract

We analyse how the spectrum of the anisotropic Maxwell system with bounded conductivity on a Lipschitz domain is approximated by domain truncation. First we prove a new non-convex enclosure for the spectrum of the Maxwell system, with weak assumptions on the geometry of the domain and none on the behaviour of the coefficients at infinity. We also establish a simple criterion for non-accumulation of eigenvalues on the imaginary axis as well as resolvent estimates. For asymptotically constant coefficients, we describe the essential spectrum and show that spectral pollution may occur only in the essential numerical range of the quadratic pencil $L_\infty(\omega)$ $=$ $\mu_\infty^{-1}$ $\mbox{curl}^2$ $-$ $\omega^2\epsilon_\infty$, acting on divergence-free vector fields. Further, every isolated spectral point of the Maxwell system lying outside the essential numerical range of the pencil $L_\infty$ and outside the part of the essential spectrum on the imaginary axis is approximated by spectral points of the Maxwell system on the truncated domains. Our analysis is based on two new abstract results on the (limiting) essential spectrum of polynomial pencils and triangular block operator matrices, which are of general interest. We believe our strategy of proof could be used to establish domain truncation spectral exactness for more general classes of non-self-adjoint differential operators and systems with non-constant coefficients.