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.

翻译：本文提出了一种新颖的PMCHWT方程稳定化策略，该策略通过利用准亥姆霍兹投影算子，解决了方程因频率和电导率引起的不稳定性问题。所得公式在整个低频范围内（包括涡流区域）均具备良好的条件数，并可应用于任意可穿透材料，涵盖从介质到导电体的各类媒质。此外，通过适当选择准亥姆霍兹分量的重标度系数，我们避免了在低频条件下因感应型和电容型磁缝激励（电路建模中常用于施加电位差的常见激励方式）而导致的典型精度损失问题。最后，基于准亥姆霍兹投影算子而非标准环-星分解的构建方式，使本公式与大多数快速求解器兼容，并能适用于多连通几何结构，且无需为寻找结构的全局环而增加额外计算开销。数值算例验证了所提出的预处理方案在单连通及多连通几何结构中的有效性。