This study concerns the derivation of the stability parameter that is deployed in the computation of the Dirac eigenvalue problem using the Petrov-Galerkin finite element method (PGFEM). The derivation assumes non-uniform mesh from the very beginning in the calculation and considers the \(\mathcal {O}(c^2)-\) correction for the stability parameter. The works done here shows that the computational results of the eigenvalues using the PGFEM with the stability parameter based on \(\mathcal {O}(c^2)-\) terms and of that based on \(\mathcal {O}(c^3)-\) terms only are almost the same. Therefore, it is not necessary to assume the \(\mathcal {O}(c^2)-\) correction for the stability parameter and it is enough to only assume the stability parameter derived upon considering the \(\mathcal {O}(c^3)-\) terms in the derivation. This minimizes the cost of the computation. Although, the derivation of the stability parameter in this work involves the three-point formulae, backward, central, and forward, together with the central five-point formula to approximate the first derivatives of the Dirac functions instead of just the backward and forward two-point formulae with non-uniform mesh, as an attempt of improvement, but this provides no significant enhancement to the computational results. Thus, there is no need to use higher order derivative approximation formulae in the derivation of the stability parameter of the Dirac eigenvalue problem. The main result of this article is that the previously derived stability parameters for computing the Dirac eigenvalues using the PGFEM are optimal and assuming a correction to them is just time consuming adding no further improvement to the approximation convergence rate.
