Spring 2020 A well-posed surface currents and charges system for electromagnetism in dielectric media
Mahadevan Ganesh, Stuart Hawkins, Cole Jeznach, Darko Volkov
J. Integral Equations Applications 32(1): 1-18 (Spring 2020). DOI: 10.1216/JIE.2020.32.1


The free space Maxwell dielectric problem can be reduced to a system of surface integral equations (SIE). A numerical formulation for the Maxwell dielectric problem using an SIE system presents two key advantages: first, the radiation condition at infinity is exactly satisfied, and second, there is no need to artificially define a truncated domain. Consequently, these SIE systems have generated much interest in physics, electrical engineering, and mathematics, and many SIE formulations have been proposed over time. In this article we introduce a new SIE formulation which is in the desirable operator form identity plus compact, is well-posed, and remains well-conditioned as the frequency tends to zero. The unknowns in the formulation are three-dimensional vector fields on the boundary of the dielectric body. The SIE studied in this paper is derived from a formulation developed in earlier work by Ganesh, Hawkins, and Volkov. Our initial formulation utilized linear constraints to obtain a uniquely solvable system for all frequencies. The new SIE introduced and analyzed in this article combines the integral equations from that previous work with new constraints. We show that the new system is in the operator form identity plus compact in a particular functional space, and we prove well-posedness at all frequencies and low-frequency stability of the new SIE.


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Mahadevan Ganesh. Stuart Hawkins. Cole Jeznach. Darko Volkov. "A well-posed surface currents and charges system for electromagnetism in dielectric media." J. Integral Equations Applications 32 (1) 1 - 18, Spring 2020. https://doi.org/10.1216/JIE.2020.32.1


Received: 2 October 2018; Revised: 14 March 2019; Accepted: 20 March 2019; Published: Spring 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07223719
MathSciNet: MR4115968
Digital Object Identifier: 10.1216/JIE.2020.32.1

Primary: 35Q61 , 45E99 , 47A53

Keywords: electromagnetic scattering , infinite dimensional pencil theory , stabilization , weakly singular surface integral equations

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


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Vol.32 • No. 1 • Spring 2020
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