Finite Element Method For Solving The Dirac Eigenvalue Problem With Linear Basis Functions

TL;DR

This paper introduces a Petrov-Galerkin finite element approach using linear basis functions to accurately solve the radial Dirac eigenvalue problem while effectively addressing the issue of spurious eigenvalues.

Contribution

It presents a novel finite element method that employs only continuous basis functions and derives a stability parameter to improve eigenvalue computation accuracy.

Findings

Successfully eliminates spurious eigenvalues in Dirac problem

Derives a mesh-dependent stability parameter $ au$

Demonstrates improved numerical stability and accuracy

Abstract

In this work we will treat the spurious eigenvalues obstacle that appears in the computation of the radial Dirac eigenvalue problem using numerical methods. The treatment of the spurious solution is based on applying Petrov-Galerkin finite element method. The significance of this work is the employment of just continuous basis functions, thus the need of a continuous function which has a continuous first derivative as a basis, as in [ALMAN,ALMANA,ALMA], is no longer required. The Petrov-Galerkin finite element method for the Dirac eigenvalue problem strongly depends on a stability parameter, $\tau$, that controls the size of the diffusion terms added to the finite element formulation for the problem. The mesh-dependent parameter $\tau$ is derived based on the given problem with the particular basis functions.