Stabilized Finite Element Method for the Radial Dirac Equation

TL;DR

This paper introduces a stabilized finite element method using SUPG to effectively eliminate spurious eigenvalues in the numerical solution of the radial Dirac eigenvalue problem, improving accuracy and reliability.

Contribution

It presents a novel FEM approach with specific trial and test spaces and develops a SUPG scheme with an explicit stability parameter to address the spurious eigenvalue issue.

Findings

Successfully removes unphysical eigenvalues from solutions

Provides a stable and accurate numerical scheme for the radial Dirac problem

Demonstrates improved convergence and reliability over previous methods

Abstract

A challenging difficulty in solving the radial Dirac eigenvalue problem numerically is the presence of spurious (unphysical) eigenvalues among the correct ones that are neither related to mathematical interpretations nor to physical explanations. Many attempts have been made and several numerical methods have been applied to solve the problem using finite element method (FEM), finite difference method (FDM), or other numerical schemes. Unfortunately most of these attempts failed to overcome the difficulty. As a FEM approach, this work can be regarded as a first promising scheme to solve the spuriousity problem completely. Our approach is based on an appropriate choice of trial and test functional spaces. We develop a Streamline Upwind Petrov-Galerkin method (SUPG) to the equation and derive an explicit stability parameter.