J. Garcke and M. Griebel.
On the computation of the eigenproblems of hydrogen and helium in
strong magnetic and electric fields with the sparse grid combination
Journal of Computational Physics, 165(2):694-716, 2000.
also as SFB 256 Preprint 670, Institut für Angewandte Mathematik,
Universität Bonn, 2000.
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We introduce the combination technique for the numerical solution of d-eigenproblems on sparse grids. Here, O(d ·(logN)d-1) different problems, each of size O(N), have to be solved independently. This is in contrast to the one problem of size O(Nd) for a conventional finite element discretization, where N denotes the number of grid points in one coordinate direction. Therefore, also higher dimensional eigenvalue problems can be treated by our sparse grid combination approach. We apply this method to solve the three-dimensional Schrödinger equation for hydrogen (one electron problem) and the six-dimensional Schrödinger equation for helium (two electron problem) in strong magnetic and electric fields.