hp-Cloud Approximation Of The Dirac Eigenvalue Problem: The Way Of Stability

TL;DR

This paper introduces an $hp$-cloud Petrov-Galerkin method to solve the radial Dirac eigenvalue problem, effectively eliminating spurious eigenvalues and ensuring stability through a carefully controlled diffusivity parameter.

Contribution

It develops a novel $hp$-cloud Petrov-Galerkin scheme with a stability parameter to address eigenvalue stability issues in the Dirac problem, including a new derivation for the diffusivity parameter.

Findings

Successfully removes spurious eigenvalues

Provides a stable and consistent numerical scheme

Applicable to generic basis functions

Abstract

We apply $hp$-cloud method to the radial Dirac eigenvalue problem. The difficulty of occurrence of spurious eigenvalues among the genuine ones in the computation is resolved. The method of treatment is based on assuming $hp$-cloud Petrov-Galerkin scheme to construct the weak formulation of the problem which adds a consistent diffusivity to the variational formulation. The size of the artificially added diffusion term is controlled by a stability parameter ($\tau$). The derivation of $\tau$ assumes the limit behavior of the eigenvalues at infinity. The parameter $\tau$ is applicable for generic basis functions. This is combined with the choice of appropriate intrinsic enrichments in the construction of the cloud shape functions.