\end{array}$$ and $$\label{jump2}
\begin{array}{l}
{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}|_F := {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_F \hbox{ with} \\ \\
{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_F := \left\{
\begin{array}{ll}
\varphi_K {\boldsymbol{n}}_K + \varphi_{K'}{\boldsymbol{n}}_{K'} & \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_I)$}\\[.1cm]
\varphi_K {\boldsymbol{n}}_\Sigma & \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Sigma$}\, .
\end{array} \right.
\end{array}$$ Similarly, we define the edge averages ${\{{\boldsymbol{v}}\}}_{{\mathcal{E}}}\in \mathbf{L}^2({\mathcal{E}}_h)$ by $${\{{\boldsymbol{v}}\}}_{{\mathcal{E}}}|_e :={\{{\boldsymbol{v}}\}}_e \hbox{ with } {\{{\boldsymbol{v}}\}}_e:=({\boldsymbol{v}}_{K_e} + {\boldsymbol{v}}_{K'_e})/2$$ where $K_e, K'_e\in {\mathcal{T}}_h^{\Omega_C}$ are such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$. We also need to define the edge jumps ${\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\in \mathbf{L}^2({\mathcal{E}}_h)$ by $${\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}|_e :=   {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_e \hbox{ with }
 {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_e := 
 \varphi_{K_e} {\boldsymbol{t}}_e + \varphi_{K'_e} {\boldsymbol{t}}'_e\, ,$$ where $K_e, K_e'$ are in this case the elements from ${\mathcal{T}}_h^{\Omega_I}$ 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 $e = T\cap T'$. Here, ${\boldsymbol{t}}_e$, ${\boldsymbol{t}}'_e$ are the tangent unit vectors along the edge $e$ given by ${\boldsymbol{t}}_e = ({\boldsymbol{n}}_\Gamma \times \boldsymbol \nu_{T})|_e$ and ${\boldsymbol{t}}_e = ({\boldsymbol{n}}_\Gamma \times \boldsymbol \nu_{T'})|_e$ where $\boldsymbol \nu_{T}$ and $\boldsymbol \nu_{T'}$ are the outward unit normal vector to $\partial T$ and $\partial T'$ respectively that lies on the tangent plane to $\Gamma$.

The DG formulation
------------------

Hereafter, given an integer 