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A Study of Space-Time Discretizations for the Dirac Equation

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.