Numerical iterative methods for the Hartree equation of helium- like system

Hartree's equation, or system of equations, was introduced by Hartree in 1928 as a nonlinear approximation of Schrodinger's linear equation, in order to replace the wave function dependent on many variables appearing in Schrodinger's equation by several functions each depending on three variables only. The mathematical study of Hartree's equation began in 1970, with Reeken's proof of the existence and unicity of the normalized pointwise positive solution of Hartree's equation for the helium atom. Hartree's equation is well adapted to numerical iterative schemes, the classical one having been introduced by Hartree himself. In this work we study several new numerical schemes for the computation of a solution of Hartree's equation for helium-like systems. One group of these schemes is based on the minimization of the Hartree energy by a decomposition coordination method via a convenient augmented Lagrangian. The other group is based on a three-stage splitting operator method for an associated evolution equation. For the discretization we use the P1 finite elements, or the finite differences for an approximation of the Laplacian. We present some numerical results and compare the methods studied.