Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T13:14:12.741Z Has data issue: false hasContentIssue false
  • English
  • Français

A Multigrid Method for Ground State Solution of Bose-Einstein Condensates

Part of: Numerical analysis: Ordinary differential equations Partial differential equations, boundary value problems Acceleration of convergence

Published online by Cambridge University Press:  16 March 2016

Hehu Xie  and
Manting Xie
Hehu Xie*
Affiliation:
LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Manting Xie
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
*Corresponding author. Email addresses:[email protected] (H. Xie), [email protected] (M. Xie)
Get access

Abstract

A multigrid method is proposed to compute the ground state solution of Bose-Einstein condensations by the finite element method based on the multilevel correction for eigenvalue problems and the multigrid method for linear boundary value problems. In this scheme, obtaining the optimal approximation for the ground state solution of Bose-Einstein condensates includes a sequence of solutions of the linear boundary value problems by the multigrid method on the multilevel meshes and some solutions of nonlinear eigenvalue problems some very low dimensional finite element space. The total computational work of this scheme can reach almost the same optimal order as solving the corresponding linear boundary value problem. Therefore, this type of multigrid scheme can improve the overall efficiency for the simulation of Bose-Einstein condensations. Some numerical experiments are provided to validate the efficiency of the proposed method.

Keywords

BECGPEeigenvalue problemmultigridmultilevel correctionfinite element method

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N25: Eigenvalue problems 65L15: Eigenvalue problems 65B99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 648 - 662
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R. A., Sobolev spaces, Academic Press, New York, 1975.Google Scholar
[2]Adhikari, S. K., Collapse of attractive Bose-Einstein condensed vortex states in a cylindrical trap, Phys. Rev. E, 65 (2002), 016703.Google Scholar
[3]Adhikari, S. K., Muruganandam, P., Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation, J. Phys. B, 35 (2002), 2831.Google Scholar
[4]Anderson, M. H., Ensher, J. R., Mattews, M. R. , Wieman, C. E. and Cornell, E. A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198–201.Google Scholar
[5]Anglin, J. R. and Ketterle, W., Bose-Einstein condensation of atomic gasses, Nature, 416 (2002), 211–218Google Scholar
[6]Bao, W. and Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condestion, Kinetic and Related Models, 6(1) (2013), 1–135.Google Scholar
[7]Bao, W. and Du, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, Siam J. Sci. Comput., 25(5) (2004), 1674–1697.Google Scholar
[8]Bao, W. and Tang, W., Ground-state solution of trapped interacting Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187 (2003), 230–254.Google Scholar
[9]Bramble, J. H., Multigrid Methods, Pitman Research Notes in Mathematics, V. 294, John Wiley and Sons, 1993.Google Scholar
[10]Bramble, J. H. and Pasciak, J. E., New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), 311–329.Google Scholar
[11]Brenner, S. and Scott, L., The Mathematical Theory of Finite Element Methods, New York: Springer-Verlag, 1994.Google Scholar
[12]Cancès, E., Chakir, R., Maday, Y., Numerical analysis of nonlinear eigenvalue problems, J. Sci. Comput., 45(1-3) (2010), 90–117.Google Scholar
[13]Cerimele, M. M., Chiofalo, M. L., Pistella, F., Succi, S., Tosi, M. P., Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: an application to trapped Bose-Einstein condensates, Phys. Rev. E, 62 (2000), 1382.Google Scholar
[14]Chien, C.-S., Huang, H.-T., Jeng, B.-W. and Li, Z.-C., Two-grid discretization schemes for nonlinear Schröinger equations, J. Comput. Appl. Math., 214 (2008), 549–571.Google Scholar
[15]Chien, C.-S., Jeng, B.-W., A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput., 27(4) (2006), 1287–1304.Google Scholar
[16]Chiofalo, M. L., Succi, S., Tosi, M. P., Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62 (2000), 7438.Google Scholar
[17]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978.Google Scholar
[18]Cornell, E. A., Very cold indeed: the nanokelvin physics of Bose-Einstein condensation J. Res. Natl Inst. Stand., 101 (1996), 419–434.Google Scholar
[19]Cornell, E. A. and Wieman, C. E., Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments, Rev. Mod. Phys., 74 (2002), 875–893.Google Scholar
[20]Dalfovo, F., Giorgini, S., Pitaevskii, L. P. and Stringari, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463–512.Google Scholar
[21]Dodd, R. J., Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates, J. Res. Natl. Inst. Stan., 101 (1996), 545.Google Scholar
[22]Edwards, M., Burnett, K., Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms, Phys. Rev. A, 51 (1995), 1382.CrossRefGoogle ScholarPubMed
[23]Griffin, A., Snoke, D. W., and Stringari, S., Bose Einstein-Condensation, Cambridge University Press, Cambridge, 1995.Google Scholar
[24]Gross, E. P., Nuovo, Cimento., 20 (1961), 454.Google Scholar
[25]Hackbush, W., Multi-grid Methods and Applications, Springer-Verlag, Berlin, 1985.Google Scholar
[26]Hau, L. V., Busch, B. D., Liu, C., Dutton, Z., Burns, M. M. and Golovchenko, J. A., Near-resonant spatial images of confined Bose-Einstein condensates in a 4-Dee magnetic bottle, Phys. Rev. A, 58 (1998), R54–57.Google Scholar
[27]Henning, P., Målqvist, A. and Peterseim, D., Two-level discretization techniques for ground state computations of Bose-Eistein condensates, SIAM J. Numer. Anal., 52(4) (2014), 1525–1550.Google Scholar
[28]Jin, S., Levermore, C. D., and McLaughlin, D. W., The semiclassical limit of the Defocusing Nonlinear Schrödinger Hierarchy, CPAM, 52 (1999), 613–654.Google Scholar
[29]Ketterle, W., Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser, Rev. Mod. Phys., 74 (2002), 1131–1151.Google Scholar
[30]Laudau, L. and Lifschitz, E., Quantum Mechanics: non-relativistic theory, Pergamon Press, New York, 1977.Google Scholar
[31]Lieb, E. H., Seiringer, R. and Yangvason, J., Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A, 61 (2000), 043602.Google Scholar
[32]Lin, Q. and Xie, H., A multi-level correction scheme for eigenvalue problems, Math. Comp., 84 (2015), 71–88.Google Scholar
[33]McCormick, S. F., ed., Multigrid Methods. SIAM Frontiers in Applied Matmematics 3. Society for Industrial and Applied Mathematics, Philadelphia, 1987.Google Scholar
[34]Schneider, B. I., Feder, D. L., Numerical approach to the ground and excited states of a Bose-Einstein condensated gas confined in a completely anisotropic trap, Phys. Rev. A, 59 (1999), 22–32.Google Scholar
[35]Xie, H., A type of multilevel method for the Steklov eigenvalue problem, IMA J. Numer. Anal., 34(2) (2014), 592–608.Google Scholar
[36]Xie, H., A type of multi-level correction method for eigenvalue problems by nonconforming finite element methods, Research Report in ICMSEC, 2012-10 (2012).Google Scholar
[37]Xie, H., A multigrid method for eigenvalue problem, J. Comput. Phys., 274 (2014), 550–561.Google Scholar
[38]Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Review, 34(4) (1992), 581–613.Google Scholar
[39]Zhou, A., An analysis of fnite-dimensional approximations for the ground state solution of Bose-Einstein condensates, Nonlinearity, 17 (2004), 541–550.Google Scholar
[40]Zienkiewicz, O. and Zhu, J., The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Internat. J. Numer. Methods Engrg., 33(7) (1992), 1331–1364.Google Scholar