A well-posed surface currents and charges system for electromagnetism in dielectric media

TL;DR

This paper introduces a new surface integral equation formulation for electromagnetism in dielectric media that is well-posed, stable at all frequencies, and maintains good conditioning as frequency approaches zero.

Contribution

The paper presents a novel SIE formulation combining previous work with new constraints, ensuring well-posedness and low-frequency stability in dielectric electromagnetism problems.

Findings

The new SIE is in the operator form identity plus compact.

The formulation is well-posed at all frequencies.

It remains well-conditioned as frequency tends to zero.

Abstract

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 some of the authors~\cite{ganesh2014all}. 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 \cite{ganesh2014all} 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.