\sigma_K$, if $F = \partial K \cap \Gamma\in {\mathcal{F}}_h^\Gamma$. We also need to define $\mathtt{s}_{\mathcal{E}}\in \prod_{e\in \mathcal{E}_h} {\mathcal{P}}_0(e)$ given by $\mathtt{s}_e = \min(\sigma_{K_e}, \sigma_{K'_e})$ where $K_e, K_e\in {\mathcal{T}}_h^{\Omega_C}$ are such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$.

We consider, for $s>1/2$, the Hilbert space $$\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) := \left\{{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C});\quad {\mathbf{curl}}_h {\boldsymbol{v}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})\right\}$$ and define on $\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$ the sesquilinear forms $$\begin{aligned}
A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), &({\boldsymbol{v}}, \varphi,m)) :=  \imath \omega \left(\mu {\boldsymbol{u}}, {\boldsymbol{v}}\right)_{{\mathcal{T}}_h^{\Omega_C}} + \left(\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}, {\mathbf{curl}}_h {\boldsymbol{v}}\right)_{{\mathcal{T}}_h^{\Omega_C}} \\ &+ \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}
+ \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{u}}, \phi, c) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}\\ &+ \mathtt{a}^{\Omega_C} \left( \mathtt{s}^{-1}_{{\mathcal{F}}} h_{{\mathcal{F}}}^{-1} {\llbracket ({\boldsymbol{u}}, \phi, c) \rrbracket}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}\, ,\end{aligned}$$ $$\begin{aligned}
A_h^{\Omega_I}(&({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)):=  \imath \omega \mu_0 (\nabla_h \phi+c \boldsymbol \rho, \nabla_h \varphi+m \boldsymbol \rho)_{{\mathcal{T}}_h^{\Omega_I}} +  \dfrac{\mathtt{a}^{\Omega_I}}{\omega \mu_0}\left(h_{{\mathcal{F}}}^{-1} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}\\
&- \imath \omega \mu_0\left({\{\nabla_h \phi+c \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}} 
- \imath \omega \mu_0\left({\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
\\ 
&- \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h} - \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}
+ \alpha \left( \mathtt{s}^{-1}_{{\mathcal{E}}} h_{{\mathcal{E}}}^{-2} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}  \right)_{{\mathcal{E}}_h},\end{aligned}$$ and let $$A_h(({\boldsymbol{u}}, p,c), ({\boldsymbol{v}}, \varphi,m)):= A_h^{\Omega_C}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)) 