(S. Larsson, S. Salomonson, N. Svanstedt, H. Almanasreh)
Many-body perturbation theory is a powerful method for high-accuracy calculations on simple atomic systems with few electrons. Highly charged ions all the way up to H-like uranium can now be studied experimentally. Numerical one-electron spectra that are complete in a discretized space can be computed and used as basis functions in perturbation calculations. The physics partners have developed numerical algorithms and have a lot of experience of such computations. Numerical bases of eigenfunctions are also used in the computation of, e.g., quantum electrodynamical corrections. High accuracy computations of this type are of interest, e.g., as a way of improving the accuracy of fundamental constants, such as the electron mass or the fine structure constant, by comparing numerical calculations with high precision measurements of energy splittings. The mathematical challenge here is to perform a mathematical analysis of the numerical methods in order to understand the difficulties and to improve the algorithms. This will be performed in the framework of finite element methods which facilitates the mathematical analysis. One interesting problem is the contamination of the numerical spectra of the Dirac equation by spurious (non-physical) states.