K} & =0, \ \forall v\in V^{p}(K), \\
-(q_h, \nabla\cdot{\mathbf{v}})_{K} + \langle\eta_{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, \ \forall {\mathbf{v}}\in\widetilde {\mathbf{V}}^{p}(K), \\
({\mathrm{i}}\omega q_h, v)_{K} - (\nabla\cdot{\mathbf{J}}_h, v)_{K} -\langle\tau_{\eta}(q_h-\eta_h), v\rangle_{\partial K} & = 0, \ \forall v\in\widetilde V^{p}(K)  .
\end{aligned}
\right.$$ One can solve the local fields element by element once the solutions for $\lambda_h$ and $\eta_h$ are obtained. In order to make the problem solvable, we need to employ global conditions $$\label{eq:globalRaw}
\left\{
\begin{aligned}
\langle{\llbracket {\mathbf{n}}\times\hat{{\mathbf{E}}}_h \rrbracket}, v\rangle_{{\mathcal{F}}_h} - \langle\lambda_h, v\rangle_{\partial\Omega} & = \langle g^{\text{inc},v}\rangle_{\partial\Omega}, \ \forall v\in M^p_h, \\
\langle{\llbracket {\mathbf{n}}\cdot\hat{{\mathbf{J}}}_h \rrbracket}, v\rangle_{{\mathcal{F}}_h^I} & = 0, \ \forall v\in\widetilde M^p_h.
\end{aligned}
\right.$$ The first relation in  weakly enforces the continuity condition for the tangential component of the electric field across any edges, and also takes into account the Silver-M[" u]{}ller absorbing boundary condition. The other global condition in Eq.  weakly enforces the continuity condition for the normal component of the current density across any edges. The additional boundary condition on the surface of the nanostructure is implicitly contained in this relation.

Substituting $\hat{{\mathbf{E}}}_h$ and $\hat{{\mathbf{J}}}_h$ in with the definitions in , we arrive at the global reduced system of equations $$\label{eq:global}
\left\{
