Spectral analysis and domain truncation for Maxwell's equations

Bögli, S., Ferraresso, Francesco ORCID: https://orcid.org/0000-0002-4399-141X, Marletta, Marco ORCID: https://orcid.org/0000-0003-1546-4046 and Tretter, C. 2023. Spectral analysis and domain truncation for Maxwell's equations. Journal de Mathématiques Pures et Appliquées 170 , pp. 96-135. 10.1016/j.matpur.2022.12.004

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Abstract

We analyse how the spectrum of the anisotropic Maxwell system with bounded conductivity σ on a Lipschitz domain Ω 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 Ω and none on the behaviour of the coefficients at infinity. We also establish a simple criterion for non-accumulation of eigenvalues at 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 , acting on divergence-free vector fields. Further, every isolated spectral point of the Maxwell system lying outside and outside the part of the essential spectrum on 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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Publisher:	Elsevier
ISSN:	0021-7824
Funders:	EPSRC, SNF
Date of First Compliant Deposit:	19 December 2022
Date of Acceptance:	13 December 2022
Last Modified:	05 Jan 2024 05:42
URI:	https://orca.cardiff.ac.uk/id/eprint/154999

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