Low-Frequency Stabilizations of the PMCHWT Equation for Dielectric and Conductive Media: On a Full-Wave Alternative to Eddy-Current Solvers

TL;DR

This paper introduces a new stabilization method for the PMCHWT equation that ensures low-frequency stability and accuracy across dielectric and conductive media, improving computational robustness and compatibility with fast solvers.

Contribution

The authors develop a quasi-Helmholtz projector-based stabilization strategy that enhances the conditioning of the PMCHWT equation at low frequencies and for complex geometries, surpassing previous methods.

Findings

Achieves well-conditioned formulations across low-frequency regimes.

Prevents accuracy loss at low frequencies with appropriate rescaling.

Compatible with fast solvers and multiply connected geometries.

Abstract

We propose here a novel stabilization strategy for the PMCHWT equation that cures its frequency and conductivity related instabilities and is obtained by leveraging quasi-Helmholtz projectors. The resulting formulation is well-conditioned in the entire low-frequency regime, including the eddy current one, and can be applied to arbitrarily penetrable materials, ranging from dielectric to conductive ones. In addition, by choosing the rescaling coefficients of the quasi-Helmholtz components appropriately, we prevent the typical loss of accuracy occurring at low frequency in the presence of inductive and capacitive type magnetic frill excitations, commonly used in circuit modeling to impose a potential difference. Finally, leveraging on quasi-Helmholtz projectors instead than on the standard Loop-Star decomposition, our formulation is also compatible with most fast solvers and is amenable to multiply connected geometries, without any computational overhead for the search for the global loops of the structure. The efficacy of the proposed preconditioning scheme when applied to both simply and multiply connected geometries is corroborated by numerical examples.