by $({\mathbf{curl}}_h {\boldsymbol{v}})|_K = {\mathbf{curl}}\, {\boldsymbol{v}}_K$, for all $K\in {\mathcal{T}}_h^{\Omega_C}$; $\nabla_h \varphi \in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_I})$ by $(\nabla_h \varphi)|_K = \nabla \varphi_K$, for all $K\in {\mathcal{T}}_h^{\Omega_I}$. We define also the averages ${\{{\boldsymbol{v}}\}}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_C})$ and ${\{\nabla_h \varphi+m\boldsymbol \rho\}}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_I})$ by $$\label{average1}
\begin{array}{l}
{\{{\boldsymbol{v}}\}}_{{\mathcal{F}}}|_F := {\{{\boldsymbol{v}}\}}_F  \hbox{ with}\\ \\
{\{{\boldsymbol{v}}\}}_F:= \left\{
\begin{array}{ll}
({\boldsymbol{v}}_K + {\boldsymbol{v}}_{K'})/2 & \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_C)$}\\[.1cm]
{\boldsymbol{v}}_K & \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Gamma$},
\end{array} \right.
\end{array}$$ and $$\label{average2}
\begin{array}{l}
 {\{\nabla_h\varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}|_F := {\{\nabla_h \varphi + m \boldsymbol \rho\}}_F \hbox{ with} \\ \\
 {\{\nabla_h \varphi + m \boldsymbol \rho\}}_F :=\left\{ \begin{array}{l} 
(\nabla \varphi_K + \nabla \varphi_{K'})/2 + m (\boldsymbol \rho_K+\boldsymbol \rho_{K'})/2 \\ \hspace{3cm} \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_I)$}\\[.1cm]
\nabla \varphi_K + m \boldsymbol \rho_K \\
\hspace{3cm} \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Sigma$}\, ,
\end{array} \right.
\end{array}$$ and the jumps ${\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_C})$ and ${\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_I})$ by $$\label{jump1}
\begin{array}{l}
{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}|_F :={\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_F \hbox{ with}
\\ \\
{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_F:=\left\{   \hspace{-.2cm}  \begin{array}{l}
{\llbracket {\boldsymbol{v}}\times {\boldsymbol{n}}\rrbracket}_F:={\boldsymbol{v}}_K \times {\boldsymbol{n}}_K + {\boldsymbol{v}}_{K'}\times {\boldsymbol{n}}_{K'} \\ \hspace{2cm} \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_C)$}\\[.1cm]
{\boldsymbol{v}}_K \times {\boldsymbol{n}}+( \nabla \varphi_{K'} +  m \boldsymbol \rho_{K'}) \times {\boldsymbol{n}}_{K'}\\ \hspace{2cm} \text{if }F= K\cap K'\in {\mathcal{F}}_h^\Gamma \text{ with } K\in {\mathcal{T}}_h^{\Omega_C}, \,K'\in {\mathcal{T}}_h^{\Omega_I}\, ,
\end{array} \right.
