Finite Element Method and Dirac eigenvalue problem Consider Dirac equation in 2D with Hamiltonian given by (arb. units)
\begin{equation}
H=-i \begin{pmatrix}
0&\partial_x-i\partial_y\\
\partial_x+i\partial_y & 0\\
\end{pmatrix}.
\end{equation}
The equation to solve, is simply
\begin{equation}
H\psi=E\psi, \ 
\psi=\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} 
\end{equation}
I expand the wave function in some basis of spatial-localized functions
\begin{equation}
\psi(x,y)=\sum_n^N \begin{pmatrix} c^1_n \\ c^2_n \end{pmatrix} \phi_n(x_n,y_n)+\sum_m^M \begin{pmatrix} b^1_m \\ b^2_m \end{pmatrix}  \phi_m(x_n,y_n)
\end{equation}
Let the functions indexed by $m$ lie on the boundary.
After plugging this form to the equation I obtain the matrix equation
\begin{equation}
H_D\psi_D=ES_D\psi_D,
\end{equation}
$H_D $ is the matrix $2(N+M) \times 2(N+M) $ (size of the problem). On the right hand side I also have overlap matrix due to non orthogonality of my basis functions.
The last thing to do is imposing the boundary condition. 
The question is, how to impose the boundary condition of the form
\begin{equation}
\psi_2=\alpha\psi_1,
\end{equation}
where $\alpha$ is some complex number. I know how to impose Dirichlet boundary condition, but I have hard time with this one. I will be glad for any ideas or any refs. Is there any name for this type of boundary ??
Greetings. 
 A: The general boundary condition for the Dirac equation is a local linear restriction on the components of the spinor wave function at the boundary,
$$\psi=M\psi,\;\;M=\begin{pmatrix}
n_z&n_x-in_y\\
n_x+in_y&-n_z
\end{pmatrix}$$
with ${\mathbf n}=(n_x,n_y,n_z)$ a unit vector. (Check that $M^2=\mathbb{1}$.)
If you are interested in the generalization of the Dirichlet boundary condition, then you will want to ensure that zero current flows through the boundary. This further restricts $M$ to vectors ${\mathbf n}$ that satisfy $({\mathbf n},{\mathbf n}_B)=0$, where ${\mathbf n}_B$ is a unit vector in the $x$-$y$ plane perpendicular to the boundary.
All of this was worked out in the context of graphene, see    arXiv:0710.2723. In that context the different boundary conditions have names: $n_x=0=n_y$ is called the "zigzag" boundary condition and $n_z=0$ is called the "mass" boundary condition. There is also something called the "armchair" boundary condition, which you need if you put the Dirac equation on a lattice.
