Stabilized Finite Element Method for the Radial Dirac Equation

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.Comment: 24 pages, 8 tables, and 1 figur

Stabilized finite element method for the radial Dirac equation

A challenging difficulty in solving the radial Dirac eigenvalue problem numerically is the presence of spurious (unphysical) eigenvalues, among the genuine 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 the finite element method (FEM), the finite difference method, 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 spuriosity problem com- pletely. Our approach is based on an appropriate choice of trial and test function spaces. We develop a Streamline Upwind Petrov–Galerkin method to the equation and derive an explicit stability parameter

Combining Quantum Electrodynamics and electron correlation

Abstract There is presently a large interest in studying highly charged ions in order to investigate the effects of quantum-electrodynamics (QED) at very strong fields. Such experiments can be performed at large accelerators, like that at GSI in Darmstadt, where the big FAIR facility is under construction. Accurate experiments on light and medium-heavy ions can also be performed by means of laser and X-ray spectroscopy. To obtain valuable information, accurate theoretical results are required to compare with. The most accurate procedures presently used for calculations on simple atomic systems are (i) all-order many-body perturbative expansion with added first-order analytical QED energy corrections, and (ii) twophoton QED calculations. These methods have the shortcoming that the combination of QED and correlational effects (beyond lowest order) is completely missing. We have developed a third procedure, which can remedy this shortcoming. Here, the energy-dependent QED effects are included directly into the atomic wave function, which is possible with the procedure that we have recently developed. The calculations are performed using the Coulomb gauge, which is most appropriate for the combined effect. Since QED effects, like the Lamb shift, have never been calculated in that gauge, this has required some development. This is now being implemented in our computational procedure, and some numerical results are presented