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On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. (English) Zbl 0979.65101
The authors propose a numerical method for Schrödinger eigenvalue problems for the helium atom in electric and magnetic fields. The high dimensionality (6 spatial dimensions) is dealt with by the use of a family of sparse grids. A grid family contains grids which are fine in only one coordinate direction and very coarse in the others, and it also contains gradations between such grids. A preconditioned conjugate gradient method is used on each grid, and a weighted average is used as the computed result. The authors point out that care must be taken in computing this average, becuase the ordering of the eigenvalues may not be consistent on the grids in the family. The method is effective if mixed derivatives are smooth.

MSC:
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35P15 Estimates of eigenvalues in context of PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
81-08 Computational methods for problems pertaining to quantum theory
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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