A justification of eddy currents model for the Maxwell equations

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.