A Justification of Eddy Currents Model for the Maxwell Equations

TLDR

The eddy currents model, derived by neglecting displacement currents in Maxwell’s equations, provides an elliptic, time‑harmonic approximation to the full system and exhibits a topology‑dependent behavior unique to Maxwell’s equations. This study shows that the eddy currents model approximates the full Maxwell system to second order in frequency only when an additional condition on the current source holds, otherwise it is only first order, and it also examines the model’s well‑posedness and time‑dependent behavior. The authors analyze the well‑posedness of the eddy currents model and investigate its time‑dependent case to establish these approximation properties. They find that the model achieves second‑order accuracy only under the source condition, otherwise it remains first order, and all results depend critically on the domain’s topological properties.

Abstract

This paper is concerned with the approximation of the Maxwell equations by the eddy currents model, which appears as a correction of the quasi-static model. The eddy currents model is obtained by neglecting the displacement currents in the Maxwell equations and exhibits an elliptic character in the time-harmonic formulation. Our main concern in this paper is to show that the eddy currents model approximates the full Maxwell system up to the second order with respect to the frequency if and only if an additional condition on the current source is fulfilled. Otherwise, it is a first-order approximation to the Maxwell equations. We also study the well-posedness of the eddy currents model and investigate the time-dependent case. All our results strongly depend on the topology properties of the domains under consideration. This dependence which is specific to Maxwell's equations does not appear for the two- or the three-dimensional Helmholtz operator.