DG methods to handle efficiently unstructured meshes with hanging nodes combined with $hp$-adaptive strategies make them well-suited for the numerical simulation of physical systems related to eddy currents.

The eddy current problem is generally written in terms of either the electric or the magnetic field, cf. [@AVbook]. These two formulations are equivalent at the continuous level but they lead to different numerical schemes. A discontinuous Galerkin method based on a time-harmonic eddy current problem written in terms of the electric field has been analyzed in the pioneering work of Perugia and Schotzau [@PS03]. For the time domain eddy current problem, Ausserhofer et al. introduced in [@ABP09] a formulation based on a magnetic vector potential and propose a numerical method that combines a DG approximation in the conductor with the usual ${\mathrm{H}}^{1}$-conforming Lagrange finite element approximation in the insulator.

Here, we are interested in imposing the magnetic field as primary unknown. The advantage of this approach rests on the reduction of the number of degrees of freedom resulting from the introduction of a scalar magnetic potential in the nonconducting medium. The global formulation of the problem consists in a ${\mathrm{H}}({\mathbf{curl}})$-elliptic problem for vector fields that are curl-free in the insulator $\Omega_I$. 