On eigenmode approximation for Dirac equations: differential forms and fractional Sobolev spaces

Abstract

We comment on the discretization of the Dirac equation using finite element spaces of differential forms. In order to treat perturbations by low order terms, such as those arising from electromagnetic fields, we develop some abstract discretization theory and provide estimates in fractional order Sobolev spaces for finite element systems. Eigenmode convergence is proved, as well as optimal convergence orders, assuming a flat background metric on a periodic domain. This research was first published in Mathematics of Computation. © American Mathematical Society.