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Frequency-Stable Full Maxwell in Electro-Quasistatic Gauge

by J. Ostrowski and R. Hiptmair

(Report number 2020-43)

Abstract
The electro-quasistatic approximation of Maxwell’s 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’s equations that consistently takes into account all electromagnetic effects. Electro-quasistatics serves as a gauge condition in this semi-decoupled 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 so-called ECE boundary conditions, which facilitate coupling with external circuit models.

Keywords: Maxwell's equations, ECE boundary conditions, quasi-static models, low-frequency breakdown, low-frequency stabilization, finite-element method

@Techreport{OH20_916,
  author = {J. Ostrowski and R. Hiptmair},
  title = {Frequency-Stable Full Maxwell in Electro-Quasistatic Gauge},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-43},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-43.pdf },
  year = {2020}
}

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