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

TL;DR

This paper develops a theoretical framework for discretizing Dirac equations with finite element differential forms, providing convergence proofs and error estimates in fractional Sobolev spaces, especially for electromagnetic perturbations.

Contribution

It introduces an abstract discretization theory for Dirac equations using differential forms and establishes eigenmode convergence with optimal rates in fractional Sobolev spaces.

Findings

Eigenmode convergence is proven.

Optimal convergence orders are established.

Error estimates are provided in 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 arizing 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.