\boldsymbol \rho) \cdot {\boldsymbol{n}}_\Gamma &\text{on $\Gamma$}\label{ModelProblem3}\\[2ex]
\int_\Gamma \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}})  \cdot (\boldsymbol \rho \times {\boldsymbol{n}}_\Gamma)&=\imath \,  \omega \mu_0 \int_{\Omega_I}  (\nabla \psi + k \boldsymbol \rho) \cdot \boldsymbol \rho\label{ModelProblem4}\\[2ex]
{\mathrm{div}}(\nabla \psi + k \boldsymbol \rho) & =  0 &\text{in $\Omega_I$}\label{ModelProblem5}\\[2ex]
\psi &= 0   &\text{on } \Sigma\label{ModelProblem6}  \, .
$$

We refer to [@AVbook Section 5] for a proof of the well-posedness of problem -.

The discrete problem {#section3}
====================

Notations
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Given a real number $r\geq 0$ and a polyhedron $\mathcal O\subset \mathbb R^d$, $(d=2,3)$, we denote the norms and seminorms of the usual Sobolev space ${\mathrm{H}}^r(\mathcal O)$ by $\|\cdot \|_{r,\mathcal O}$ and $|\cdot|_{r,\mathcal O}$ respectively (cf. [@McLean]). We use the convention ${\mathrm{L}}^2(\mathcal O):= {\mathrm{H}}^0(\mathcal O)$ and ${\bf L}^2(\mathcal O):= [{\mathrm{L}}^2(\mathcal O)]^3$. We recall that, for any $t \in [-1,\: 1 ]$, the spaces ${\mathrm{H}}^{t}(\partial \mathcal O)$ have an intrinsic definition (by localization) on the Lipschitz surface $\partial \mathcal O$ due to their invariance under Lipschitz coordinate transformations. Moreover, for all $0< t\leq 1$, ${\mathrm{H}}^{-t}(\partial\mathcal O)$ is the dual of ${\mathrm{H}}^{t}(\partial\mathcal O)$ with respect to the pivot space ${\mathrm{L}}^2(\partial \mathcal{O})$. Finally we consider $\mathbf{H}({\mathbf{curl}}, \mathcal O):=\{ {\boldsymbol{v}}\in {\mathrm{L}}^2(\mathcal O)^3 \, : \, {\mathbf{curl}}\, {\boldsymbol{v}}\in {\mathrm{L}}^2(\mathcal O)^3\}$ and endow it 