valid whatever DG scheme is adopted.

Hybridizable DG implementation of the electrodynamical equations  {#subsec:hdg}
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In Sec. \[Sec:promiseHDG\] we mentioned that hybridized DG methods have advantages as compared to the classic DG schemes, and here we discuss the hybridized approach in more detail. Unlike in the above classic DG formulations where the numerical traces directly couple the values from the elements on both sides of the edges, in a HDG formulation the numerical traces are defined through hybrid variables. Introducing two hybrid variables $\lambda_h$ and $\eta_h$ which live only on the boundaries of the elements, we define the numerical traces by $$\label{eq:numTrac}
\left\{
\begin{aligned}
\hat{H}_h & = \lambda_h, \\
\hat{{\mathbf{E}}}_h & = {\mathbf{E}}_h + \tau_{\lambda}(\lambda_h-H_h){\mathbf{t}}, \\
\hat{q}_h & = \eta_h, \\
\hat{{\mathbf{J}}}_h & = {\mathbf{J}}+ \tau_{\eta}(q_h-\eta_h){\mathbf{n}},
\end{aligned}
\right.$$ where $\tau_{\lambda}$ and $\tau_{\eta}$ are two stabilization parameters. Replacing the numerical traces in with the expressions in and applying Green’s formulas to the first and fourth equations in , we obtain the local formulation of the HDG method as $$\label{eq:MaxHydIntGreen2}
\left\{
\begin{aligned}
({\mathrm{i}}\omega \varepsilon_0{\mathbf{E}}_h, {\mathbf{v}})_{K} + (H_h, \nabla\times {\mathbf{v}})_{K} - \langle\lambda_{h},{\mathbf{n}}\times{\mathbf{v}}\rangle_{\partial K} - ({\mathbf{J}}_h, {\mathbf{v}})_{K} & = 0, \ \forall {\mathbf{v}}\in {\mathbf{V}}^{p}(K), \\
({\mathrm{i}}\omega\mu_0 H_h, v)_{K} - (\nabla\times{\mathbf{E}}_h, v)_{K} + \langle\tau_{\lambda}(H_h-\lambda_h), v\rangle_{\partial 