have the same additional boundary condition ${\mathbf{n}}\cdot{\mathbf{J}}=0$ on $\partial\Omega_S$ as in the NHD model.

Specification to 2D TM mode
---------------------------

Now we can couple Maxwell’s equation with for the NHD model, or similarly with for the GNOR model. We will compute light extinction by infinitely long nanowires. We take the wire axes along the $z$-direction and consider TM-polarized incident light, i.e. polarized in the $(x,y)$-plane. In this 2D setting, we can define ${\mathbf{E}}= (E_x, E_y)^T$ to be a vector and $H = H_z$ a scalar function. Coupling the time-harmonic Maxwell’s equations and hydrodynamic Drude model , we have in 2D $$\label{eq:hyd}
\left\{
\begin{aligned}
\nabla\times H & =  -{\mathrm{i}}\omega\varepsilon_0{\mathbf{E}}+{\mathbf{J}}, \text{ in } \Omega, \\
\nabla\times{\mathbf{E}}& = {\mathrm{i}}\omega\mu_0 H, \text{ in } \Omega,\\
\nabla(\nabla\cdot{\mathbf{J}}) + \frac{\omega(\omega+{\mathrm{i}}\gamma)}{\beta^2}{\mathbf{J}}& =\frac{{\mathrm{i}}\omega\omega^2_p\varepsilon_0}{\beta^2}{\mathbf{E}}, \text{ in } \Omega_S.
\end{aligned}
\right.$$ If the Silver-M[ü]{}ller boundary condition (first-order absorbing boundary condition) [@Stupfel1994] is applied on the boundary $\partial\Omega$ of the computational domain, then we have the boundary conditions $$\label{eq:bc}
\left\{
\begin{aligned}
{\mathbf{n}}\times{\mathbf{E}}-H & = {\mathbf{n}}\times{\mathbf{E}}^{\text{inc}}-H^{\text{inc}} = g^{\text{inc}}, \text{ on } \partial\Omega, \\
{\mathbf{n}}\cdot{\mathbf{J}}& =0, \text{ on } \partial\Omega_S,
\end{aligned}
\right.$$ where ${\mathbf{E}}^{\text{inc}}$ and $H^{\text{inc}}$ stand for the electromagnetic fields of the incoming light.

HDG formulations of nonlocal optical response models  {#sec:hdg}
