of the DOFs of the additional hybrid variables only. In this way, the number of globally coupled DOFs is greatly reduced as compared to the classic DG method. In a recent study [@Yakovlev:2016], the authors showed that HDG methods outperform FEM in many cases.

Computational concepts and notations
------------------------------------

In order to give a clear presentation of the HDG method, here we introduce some computational concepts and notations. We divide the computational domain $\Omega$ into triangle elements. The union of all the triangles is denoted by ${\mathcal{T}}_h$. By ${\mathcal{F}}_h$ we denote the union of all edges of ${\mathcal{T}}_h$. Furthermore, ${\mathcal{F}}^I_h$ stands for the union of all the edges associated with the nanostructure. For an edge associated with two elements $F =\overline{K^+}\cap\overline{K^-}\in{\mathcal{F}}_h$, let $({\mathbf{v}}^\pm, v^\pm)$ be the *traces* of $({\mathbf{v}}, v)$ on $F$ from the interior of $K^{\pm}$, see Fig. \[fig:interEdge\], where we use the term *trace* to denote the restriction of a function on the boundaries of the elements [@Arnold2002]. Note that from now on ${\mathbf{v}}$ is used to describe a general vector function instead of velocity.

![Two neighboring discretization elements (here: triangles) within the computational domain. An edge $F$ is shared by two elements $K^+$ and $K^-$. The outward 