A Spacetime Finite Elements Method to Solve the Dirac Equation

TL;DR

This paper introduces a fully implicit space-time finite element method for solving the Dirac equation across multiple dimensions, demonstrating its effectiveness through various quantum phenomena simulations and parallel performance analysis.

Contribution

It presents a novel space-time finite element approach for the Dirac equation, utilizing PETSc/Tao for efficient linear system solutions and parallel computation.

Findings

Successfully simulated plane wave solutions, Zitterbewegung, and Klein paradox.

Demonstrated good parallel scalability of the method.

Validated the accuracy and efficiency of the approach.

Abstract

In this work, a fully implicit numerical approach based on space-time finite element method is presented to solve the Dirac equation in 1 (space) + 1 (time), 2 + 1, and 3 + 1 dimensions. We utilize PETSc/Tao library to implement our linear system and for using Krylov subspace based solvers such as GMRES. We demonstrate our method by analyzing several different cases including plane wave solution, Zitterbewegung, and Klein paradox. Parallel performance of this implementation is also presented.