  We study several numerical discretization techniques for the one-space plus
one-time dimensional Dirac equation, including finite difference and space-time
finite element methods. Two finite difference schemes and several space-time
finite elements function spaces are analyzed with respect to known analytic
solutions. Further we propose a finite element discretization along the
equations' characteristic lines, creating diamond-shaped elements in the
space-time plane. We show that the diamond shaped elements allow for physically
intuitive boundary conditions, improve numerical efficiency, and reduce the
overall error of the computed solution as compared to the other finite
difference and space-time finite element discretizations studied in this paper.
