\left\{
\begin{aligned}
\hat{H}_h & ={\left\{ H_h \right\}} + \alpha_{E}{\llbracket {\mathbf{n}}\times{\mathbf{E}}_h \rrbracket}, \\
{\mathbf{n}}\times\hat{{\mathbf{E}}}_h & = {\left\{ {\mathbf{n}}\times{\mathbf{E}}_h \right\}}+\alpha_{H}{\llbracket H_h \rrbracket},\\
\hat{q}_h & = {\left\{ q_h \right\}}+\alpha_J{\llbracket {\mathbf{n}}\cdot{\mathbf{J}}_h \rrbracket},\\
{\mathbf{n}}\cdot\hat{{\mathbf{J}}}_h & ={\left\{ {\mathbf{n}}\cdot{\mathbf{J}}\right\}}+\alpha_q{\llbracket q_h \rrbracket}.
\end{aligned}
\right.$$ In these definitions there is still freedom to choose values for the $\alpha$ parameters, and this corresponds to different DG schemes: by setting $\alpha_{E}=\alpha_{H}=\alpha_{J}=\alpha_{q}=0$, one obtains the *centered flux* DG scheme. With $\alpha_{E}=\alpha_{H}=\alpha_{J}=\alpha_{q}=1$, one obtains the *upwind flux* DG scheme [@ElBouajaji2013]. For more validated DG schemes, we refer the interested readers to Ref. [@Arnold2002]. Having defined the numerical traces, we finally form a global system of linear equations involving all the DOFs on all the elements $$\label{eq:MaxHydIntGreen}
\left\{
\begin{aligned}
({\mathrm{i}}\omega \varepsilon_0{\mathbf{E}}_h, {\mathbf{v}})_{K} + (H_h, \nabla\times {\mathbf{v}})_{K} -  \langle\hat{H}_{h},{\mathbf{n}}\times{\mathbf{v}}\rangle_{\partial K} - ({\mathbf{J}}_h, {\mathbf{v}})_{K} & = 0 \quad  \forall {\mathbf{v}}\in {\mathbf{V}}^{p}(K), \\
({\mathrm{i}}\omega\mu_0 H_h, v)_{K} - ({\mathbf{E}}_h, \nabla\times v)_{K} - \langle{\mathbf{n}}\times\hat{{\mathbf{E}}}_{h}, v\rangle_{\partial K} & =0 \quad \forall v\in V^{p}(K), \\
-(q_h, \nabla\cdot{\mathbf{v}})_{K} + \langle \hat{q}_{h}, {\mathbf{n}}\cdot{\mathbf{v}}\rangle_{\partial K} +  \Big(\frac{\gamma-{\mathrm{i}}\omega}{\beta^2}{\mathbf{J}}_h, {\mathbf{v}}\Big)_{K} - \Big(\frac{\omega_p^2\varepsilon_0}{\beta^2}{\mathbf{E}}_h, {\mathbf{v}}\Big)_{K} & = 0 \quad \forall {\mathbf{v}}\in \widetilde{\mathbf{V}}^{p}(K), \\
({\mathrm{i}}\omega q_h, v)_{K} + ({\mathbf{J}}_h, \nabla v)_{K} -\langle{\mathbf{n}}\cdot\hat{{\mathbf{J}}}_{h}, v\rangle_{\partial K} & = 0 \quad \forall v\in\widetilde V^{p}(K),
\end{aligned}
\right.$$ which are coupled equations that are 