Finite element method for solving the Dirac eigenvalue problem with linear basis functions

Highlights: • In this work we will provide a stable finite element computation of the eigenvalues of the Dirac operator. • The finite element scheme we provide here based on applying the stream line upwind Petrov–Galerkin instead of the usual Galerkin FEM to produce diffusivity controlled by a stability parameter derived for the specific problem. • The stability parameter controls the size of the added diffusion terms to the usual Galerkin formulation of the problem and its derivation is particular for finite element formulation of the Dirac eigenvalue problem with linear basis functions. -- 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 [2], [3], is no longer required. The Petrov–Galerkin finite element method for the Dirac eigenvalue problem strongly depends on a stability parameter, τ, that controls the size of the diffusion terms added to the finite element formulation for the problem. The mesh-dependent parameter τ is derived based on the given problem with the particular basis functions.