Frequency-Stable Full Maxwell In Electro-Quasistatic Gauge

The electro-quasistatic approximation of Maxwell equations is commonly used to model coupled resistive/capacitive phenomena at low frequencies. It neglects induction and becomes unstable in the stationary limit. We introduce a stabilization that prevents this low-frequency breakdown. It results in a system for the electric scalar potential that can be used for electro-quasistatics, electrostatics, as well as DC conduction. Our main finding is that the electro-quasistatic fields can be corrected for magnetic/inductive phenomena at any frequency in a second step. The combined field from both steps is a solution of the full Maxwell equations that consistently takes into account all electromagnetic effects. Electro-quasistatics serves as a gauge condition in this semidecoupled procedure to calculate the electromagnetic potentials. We derive frequency-stable weak variational formulations for both steps that (i) immediately lend themselves to finite-element Galerkin discretization, and (ii) can be equipped with the so-called electric circuit element (ECE) boundary conditions, which facilitate coupling with external circuit models.