for scalar functions $u$ and $v$ in $L^2(D)$, the inner product is defined as $(u,v)_D = {\displaystyle}\int_D u  \overline v \,{\rm d}x$ provided $D$ is a domain in $\mathbb{R}^2$. Finally we define the edge overlap $\langle  u,v\rangle_F ={\displaystyle}\int_F u \overline v \, {\rm d}s$, where $F$ is a specific edge. Accordingly, we can define the total edge overlap for the whole triangulation or for relevant subsets of edges. Important cases are $$\langle\cdot,\cdot\rangle_{{\mathcal{F}}_h}  =
    \sum_{F\in{\mathcal{F}}_h}\langle\cdot,\cdot\rangle_F, \qquad
     \langle\cdot,\cdot\rangle_{\partial\Omega}  =
    \sum_{F\in{\mathcal{F}}_h\cap\partial\Omega}\langle\cdot,\cdot\rangle_F,
     \qquad
    \langle\cdot,\cdot\rangle_{{\mathcal{F}}^I_h}  =
    \sum_{F\in{\mathcal{F}}^I_h}\langle\cdot,\cdot\rangle_F.$$ denoting, respectively, the total edge overlap on the computational domain, the cumulative edge overlap on the absorbing boundary of the computational domain, and finally the cumulative edge overlap on the nanostructure.

DG formulation of the coupled electrodynamical equations  {#subsec:dg}
---------------------------------------------------------

We begin the construction of a DG implementation of the hydrodynamic Drude model by rewriting the coupled electrodynamical equations  into a system of first-order equations $$\label{eq:MaxHyd}
\left\{
\begin{aligned}
{\mathrm{i}}\omega \varepsilon_0{\mathbf{E}}+ \nabla\times H -{\mathbf{J}}&= 0 \qquad \text{ in } \Omega, \\
{\mathrm{i}}\omega\mu_0 H - \nabla\times{\mathbf{E}}& =0 \qquad \text{ in } \Omega, \\
\nabla q +\frac{ \gamma-{\mathrm{i}}\omega}{\beta^2}{\mathbf{J}}- \frac{\omega_p^2\varepsilon_0}{\beta^2}{\mathbf{E}}& = 0 \qquad \text{ in } \Omega_S, \\
{\mathrm{i}}\omega q - \nabla\cdot{\mathbf{J}}& = 0 \qquad \text{ in } \Omega_S,
\end{aligned}
\right.$$ where 