Eddy current approximation in Maxwell obstacle problems

Abstract

This paper analyzes the mathematical modeling of the transient eddy current approximation in the Maxwell obstacle problem. Here, the medium is assumed to be solely open, containing conducting and non-conducting materials with certain properties of anisotropy and non-smoothness. The proposed evolutionary PDE model preserves the Faraday law and excludes the displacement current from the governing Ampère–Maxwell variational inequality (VI). Our study strives to justify this model and delivers two main results: Global well-posedness of the model and its quantitative precision by uniform a priori estimates. The latter result yields an explicit bound for the smallness condition on the ratio between the electric permittivity and the electric conductivity in the region where the displacement current is disregarded. Below this threshold, the eddy current solution provides the desired reasonable approximation and justifies the proposed model.