Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations

TL;DR

This paper introduces a new integral equation approach for solving time harmonic Maxwell equations in unbounded domains, avoiding common numerical issues and providing insights into special solution families.

Contribution

It develops a novel representation for outgoing solutions, leading to a stable integral equation method that handles complex boundary conditions and extends classical harmonic fields to non-zero frequencies.

Findings

New integral equation representation for Maxwell scattering

Avoids spurious resonances and low frequency breakdown

Establishes existence of k-Neumann fields for non-simply connected domains

Abstract

In this paper, we develop a new representation for outgoing solutions to the time harmonic Maxwell equations in unbounded domains in $\bbR^3.$ This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of non-trivial families of time harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as $k$-Neumann fields, since they generalize, to non-zero wave numbers, the classical harmonic Neumann fields. The existence of $k$-harmonic fields was established earlier by Kress.