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 c urrents in the Maxwell equations and exhibits an elliptic character in the time-harmonic formulation. Our main concern in this paper is to show that t he eddy currents model approximates the full Maxwell system up to the secon d order with respect to the frequency if and only if an additional conditio n on the current source is fulfilled. Otherwise, it is a first-order approx imation to the Maxwell equations. We also study the well-posedness of the e ddy currents model and investigate the time-dependent case. All our results strongly depend on the topology properties of the domains under considerat ion. This dependence which is specific to Maxwell's equations does not appe ar for the two- or the three-dimensional Helmholtz operator.