Comparison of integral equations for the Maxwell transmission problem with general permittivities

TL;DR

This paper compares two integral equations for the Maxwell transmission problem, demonstrating that one formulation is free from false eigenwavenumbers and false essential spectrum, and is numerically competitive for non-magnetic materials with general permittivities.

Contribution

It identifies a superior integral equation formulation that avoids false eigenwavenumbers and false essential spectrum, improving numerical stability for Maxwell transmission problems.

Findings

One integral equation is free from false eigenwavenumbers for all passive materials.

The same integral equation is free from false essential spectrum on non-smooth surfaces.

The integral equation is numerically competitive despite using eight scalar surface densities.

Abstract

Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.