To this end, we introduce a bounded domain $D$ containing in its interior $\overline{\Omega}_{C}$ and whose connected boundary $\Sigma= \partial D$ is located at a large enough distance from the conductor $\Omega_{C}$. The bounded domain $\Omega_I:= D\setminus \overline \Omega_C$ represents then the nonconducting region of the computational domain $D$.

Under our assumptions, the first de Rham cohomology group $\mathcal H^1(\Omega_I)$ of $\Omega_{I}$, namely, the space of curl-free vector fields that are not gradients, has dimension one. If we assume that $\Omega_I$ is a polyhedral domain endowed with a tetrahedral mesh, one can use the technique given in [@BRS02] for the explicit construction of a piecewise-linear vector field $\boldsymbol \rho$ spanning $\mathcal H^1(\Omega_I)$ and satisfying $\boldsymbol \rho \times {\bf n}_\Sigma = {\bf 0}$ on $\Sigma$, where ${\bf n}_\Sigma$ denotes the outward unit normal vector to $\Sigma$. For an alternative construction of $\boldsymbol \rho$ see Alonso Rodríguez et al. [@ABGV13].

The eddy current problem formulated in terms of the magnetic field ${\boldsymbol{h}}$ and the scalar magnetic potential $\psi$ reads as follows: $$\label{model0}
\begin{array}{rcll}
\imath \omega \mu {\boldsymbol{h}}+ {\mathbf{curl}}\, {\boldsymbol{e}}&=& \boldsymbol 0         &\text{in $D$}\\[2ex]
 {\boldsymbol{e}}&=& \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}})         &\text{in $\Omega_C$}\\[2ex]
        {\boldsymbol{h}}&=& \nabla \psi + k \boldsymbol \rho  &\text{in $\Omega_I$}\\[2ex]
        \psi &=& 