2 | \text{Re} \left( 
 {\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}} | \leq
2{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi+ m \boldsymbol \rho\}}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}  {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}\\[.1cm]
\qquad \qquad \leq 2 C_{\Omega_I} {\lVert\nabla_h \varphi+m \boldsymbol \rho\rVert}_{0,\Omega} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket  \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\qquad\qquad \leq \frac{1}{2} {\lVert\nabla_h \varphi + m \boldsymbol \rho\rVert}_{0,\Omega}^2 + 4 C_{\Omega_I}^2 {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket  \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}^2.  
\end{array}$$ Finally, using we have that $$\label{elip3}
\begin{array}{l}
2 |\text{Re}  \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \overline \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h}|\leq 
2 {\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} 
{\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket  \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}  \\[.1cm]
\qquad \leq  
2 C_\Gamma  {\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}^2_{0,\Omega_C}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket  \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm] 
\qquad\leq  \frac{1}{4}  
{\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 + 4 C_{\Omega_C}^2 {\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket  \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}^2.
\end{array}$$ Combining with - and choosing $\alpha_0 = 1/2 + 4 C_\Omega^2 + 4C_{\Omega_I}^2$ we obtain .

We are now in a position to prove the ${\lVert\cdot\rVert}$-stability of the DG scheme .

\[LM\] Assume that $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ and $\min(\mathtt{a}^\Omega, \mathtt{a}^{\Omega_I}, \alpha)\geq \alpha_0$. Then, there exits a unique $({\boldsymbol{h}}_{h},\psi_h,k_h) \in \mathbf{X}_h\times V_h\times {\mathbb C}$ solution of Problem . Moreover if $({\boldsymbol{h}}, \psi,k)\in [\mathbf{H}({\mathbf{curl}},\Omega)\times {\mathrm{H}}^1(\Omega_I)\times {\mathbb C}] \cap  [\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}]$ is the solution to - then $$\label{Cea}
{\lVert({\boldsymbol{h}}- {\boldsymbol{h}}_{h}, \psi - \psi_h, k -k_h)\rVert} \leq (1 