{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{F}\\
- \imath \omega \mu_{0} \sum_{T\in {\mathcal{F}}^{\Gamma}_{h}}\int_{F} (\nabla \psi + k \boldsymbol \rho)
\cdot \varphi{\boldsymbol{n}}_{\Gamma} + \imath \omega \mu_{0} \sum_{T\in {\mathcal{F}}^{\Sigma}_{h}}\int_{F} (\nabla \psi + k \boldsymbol \rho) \cdot \varphi{\boldsymbol{n}}_{\Sigma} + m \int_{\Gamma} {\boldsymbol{e}}\cdot (\boldsymbol \rho \times {\boldsymbol{n}}_{\Gamma}).\end{gathered}$$ Substituting back and in we obtain $$\begin{gathered}
\label{diff1}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = -
\sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T {\boldsymbol{e}}\cdot {\mathbf{curl}}_T \varphi\\ 
- \imath \omega \mu_0 \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T \nabla (\psi+ k \boldsymbol \rho) \cdot \varphi {\boldsymbol{n}}_{\Gamma} - \left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.\end{gathered}$$ Finally, using the integration by parts formula $$\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} {\boldsymbol{e}}\cdot {\mathbf{curl}}_T \varphi =\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} (\text{curl}_T {\boldsymbol{e}})  \varphi - \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_{\partial T} {\boldsymbol{e}}\cdot \varphi{\boldsymbol{t}}_{\partial T}
=
\int_{\Gamma} (\text{curl}_\Gamma {\boldsymbol{e}})  \varphi -
\left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h},$$ we deduce from that $$\begin{gathered}
\label{diff1+}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = -
\int_{\Gamma} (\text{curl}_\Gamma {\boldsymbol{e}})  \varphi\\ 
- \imath \omega \mu_0 \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T \nabla (\psi+ k \boldsymbol \rho) \cdot \varphi {\boldsymbol{n}}_{\Gamma}.\end{gathered}$$ and the result follows from the identity $\text{curl}_\Gamma {\boldsymbol{e}}= {\mathbf{curl}}{\boldsymbol{e}}\cdot {\boldsymbol{n}}$, equation and the transmission condition .

Convergence analysis of the DG-FEM formulation {#section4}
==============================================

The aim of this Section is to prove that the DG-FEM formulation is stable in the DG-norm defined on 