zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Ackermann, J.; Erdmann, B.; Roitzsch, R., A self-adaptive multilevel finite element method for the stationary Schrödinger equation in three space dimensions, J. chem. phys., 101, 7643, (1994)
[2] Ackermann, J.; Roitzsch, R., A two-dimensional multilevel adaptive finite element method for the time-dependent Schrödinger equation, Chem. phys. lett., 214, 109, (1993)
[3] Babuska, I.; Rheinboldt, W., Error estimates for adptive finite element computations, SIAM J. numer. anal., 15, 736, (1978) · Zbl 0398.65069
[4] Baker, J.D.; Freund, D.E.; Hill, R.N.; Morgan, J.D., Radius of convergence and analytic behavior of the 1/z expansion, Phys. rev. A, 41, 1247, (1990)
[5] Balder, R., Adaptive verfahren für elliptische und parabolische differentialgleichungen auf dünnen gittern, (1994)
[6] G. Baszenski, Nth order polynomial spline blending, in, Multivariate Approximation III, K. Zeller and W. Schemp, Birkäuser, Basel, 1985. · Zbl 0568.41008
[7] Becken, W.; Schmelcher, P.; Diakonos, F.K., Helium in strong magnetic fields, J. phys. B, 32, 1557, (1999)
[8] Braun, M.; Schweizer, W.; Elster, H., Hyperspherical close coupling calculations for helium in a strong magnetic field, Phys. rev. A, 57, 3739, (1998)
[9] Bungartz, H.-J., Dünne gitter und deren anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-gleichung, (1992)
[10] Bungartz, H.-J., Finite elements of higher order on sparse grids, (1998)
[11] Bungartz, H.-J.; Dornseifer, T.; Zenger, C., Tensor product approximation spaces for the efficient numerical solution of partial differential equations, Proc. int. workshop on scientific computations, konya, (1996)
[12] Bungartz, H.-J.; Griebel, M.; Röschke, D.; Zenger, C., Pointwise convergence of the combination technique for the Laplace equation, East-west J. numer. math., 2, 21, (1994) · Zbl 0807.65105
[13] Bungartz, H.-J.; Griebel, M.; Rüde, U., Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems, Comput. methods appl. mech. eng., 116, 243, (1994) · Zbl 0824.65104
[14] E. Bylaska, et al., Scalable parallel numerical methods and software tools for material design, in, Proc. 7th SIAM Conf. on Parallel Processing for Scientific Computing, February 1995, San Francisco, CA, 1995. · Zbl 0836.65132
[15] Faßbinder, P.; Schweizer, W., The hydrogen atom in very strong magnetic and electric fields, Phys. rev. A, 53, 2135, (1996)
[16] Frank, K.; Heinrich, S.; Pereverzev, S., Information complexity of multivariate Fredholm equations in Sobolev classes, J. complexity, 12, 17, (1996) · Zbl 0858.65131
[17] Garcke, J., Berechnung von eigenwerten der stationären schrödingergleichung mit der kombinationstechnik, (1998)
[18] Gerstner, T.; Griebel, M., Numerical integration using sparse grids, Numer. algorithms, 18, 209, (1998) · Zbl 0921.65022
[19] Griebel, M., The combination technique for the sparse grid solution of PDEs on multiprocessor machines, Parallel process. lett., 2, 61, (1992)
[20] Griebel, M., Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences, Computing, 61, 151, (1998) · Zbl 0918.65078
[21] M. Griebel, W. Huber, T. Störtkuhl, and, C. Zenger, On the parallel solution of 3D PDEs on a network of workstations and on vector computers, in, Parallel Computer Architectures: Theory, Hardware, Software, Applications, Lecture Notes in Computer Science, edited by, A. Bode and M. Dal, Cin, Springer-Verlag, Berlin/New York, 1993, Vol, 732, p, 276.
[22] Griebel, M.; Knapek, S., Optimized approximation spaces for operator equations, Constr. approx., 16, 525-540, (2000) · Zbl 0969.65107
[23] Griebel, M.; Oswald, O.; Schiekofer, T., Sparse grids for boundary integral equations, Numer. math., 83, 279, (1999) · Zbl 0935.65131
[24] M. Griebel, M. Schneider, and, C. Zenger, A combination technique for the solution of sparse grid problems, in, Iterative Methods in Linear Algebra, edited by, R. Beauwens and P. de Groen, Elsevier, Amsterdam/New York, 1992. · Zbl 0785.65101
[25] Griebel, M.; Thurner, V., The efficient solution of fluid dynamics problems by the combination technique, Int. J. numer. methods heat fluid flow, 5, 254, (1995)
[26] Grindstein, F.F.; Rabitz, H.; Askar, A., The multigrid method for accelerated solution of the discretized Schrödinger equation, J. comput. phys., 51, 423, (1983) · Zbl 0582.65087
[27] Hilgenfeldt, S., Numerische Lösung der stationären schrödingergleichung mit finite-element-methoden auf dünnen gittern, (1994)
[28] Hilgenfeldt, S.; Balder, S.; Zenger, C., Sparse grids: applications to multi-dimensional Schrödinger problems, (1995)
[29] Huber, W., Turbulenzsimulation mit der kombinationsmethode auf workstation-netzen und parallelrechnern, (1996)
[30] Jones, M.D.; Ortiz, G.; Ceperley, D.M., Hartree-Fock studies of atoms in strong magnetic fields, Phys. rev. A, 54, 219, (1996)
[31] Knapek, S., Approximation und kompression mit tensorprodukt-multiskalen-approximationsräumen, (2000)
[32] Kravchenko, Yu.P.; Liberman, M.A.; Johansson, B., Exact solution for a hydrogen atom in a magnetic field of arbitrary strength, Phys. rev. A, 54, 287, (1996)
[33] Larsen, D.M., Variational studies of bound states of the H−ion in a magnetic field, Phys. rev. B, 20, 5217, (1979)
[34] Lippert, R.; Arias, T.; Edelman, A., Multiscale computations with interpolating wavelets, J. comput. phys., 140, 278, (1998) · Zbl 0927.65130
[35] Longsine, D.E.; McCormick, S.F., Simultaneous Rayleigh-quotient minimization method for ax =λbx, Linear algebra appl., 34, 195, (1980) · Zbl 0475.65021
[36] Maischak, M., Hp-methoden für randintegralgleichungen bei 3D-problemen, theorie und implementierung, (1995)
[37] Pflaum, C., Diskretisierung elliptischer differentialgleichungen mit dünnen gittern, (1996)
[38] Pflaum, C.; Zhou, A., Error analysis of the combination technique, Numer. math., 84, 327, (1999) · Zbl 0942.65122
[39] Ruder, H.; Wunner, G.; Herold, H.; Geyer, F., Atoms in strong magnetic fields, (1994)
[40] Schiekofer, T., Die methode der finiten differenzen auf dünnen gittern zur Lösung elliptischer und parabolischer pdes, (1998)
[41] Schmelcher, P.; Schweizer, W., Atoms and molecules in strong external fields, (1998)
[42] Scrinzi, A., Helium in a cylindrically symmetric field, J. phys. B, 29, 6055, (1996)
[43] Sickel, W.; Sprengel, F., Interpolation on sparse grids and tensor products of nikol’skij-Besov spaces, J. comput. anal. appl., 1, 261, (1999) · Zbl 0945.41002
[44] Smolyak, S., Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. akad. nauk SSSR, 4, 240, (1963) · Zbl 0202.39901
[45] Temlyakov, V., Approximation of functions with bounded mixed derivative, Proc. Steklov inst. math., 1, (1989) · Zbl 0668.41024
[46] Thurner, G.; Herold, H.; Ruder, H.; Schlicht, G.; Wunner, G., Note on binding energies of helium-like systems in magnetic fields, Phys. lett., 89A, 133, (1982)
[47] Wilkinson, J.H., The algebraic eigenvalue problem, (1965) · Zbl 0258.65037
[48] C. Zenger, Sparse grids, in, Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, edited by, W. Hackbusch, Vieweg-Verlag, Wiesbaden, 1991.
[49] G. Zumbusch, A sparse grid PDE solver, in, Advances in Software Tools for Scientific Computing, Lecture Notes in Computational Science and Engineering, edited by, H. P. Langtangen, A. M. Bruaset, and E. Quak, Springer-Verlag, Berlin/New York, 2000, Vol, 10, p, 133.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.