On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique

We introduce the combination technique for the numerical solution of d-dime nsional eigenproblems on sparse grids. Hen, O(d.(log N)(d-1)) different pro blems, each of size O(N), have to be solved independently. This is in contr ast to the one problem of size O(Nd) for a conventional finite element disc retization, where N denotes the number of grid points in one coordinate dir ection. Therefore, also higher dimensional eigenvalue problems can be treat ed by our sparse grid combination approach. We apply this method to solve t he three-dimensional Schrodinger equation for hydrogen (one-electron proble m) and the six-dimensional Schrodinger equation for helium (two-electron pr oblem) in strong magnetic and electric fields. (C) 2000 Academic Press.