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

Abstract

We propose here a novel stabilization strategy for the Poggio–Miller–Chang–Harrington–Wu–Tsai (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, because it relies on quasi-Helmholtz projectors instead of the standard loop-star decomposition, our formulation can be used for modeling multiply connected geometries, without incurring the computational overhead caused by the search for the global loops of the structure, while also being compatible with most fast solvers. The efficacy of the proposed preconditioning scheme when applied to both simply and multiply connected geometries is corroborated by numerical examples.