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An Efficient Adaptive Mesh Redistribution Method for Nonlinear Eigenvalue Problems in Bose–Einstein Condensates

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Abstract

We design a multilevel correction type of adaptive finite element method based on the moving mesh technique for solving nonlinear eigenvalue problems. In this paper, we take the ground state of Bose–Einstein condensates as the example of a nonlinear eigenvalue problem to show the solving process. For this aim, we propose a non-nested augmented subspace method for the nonlinear eigenvalue problems since the sequence of finite element spaces generated by the r-adaptive method has non-nested property. The new method proposed in this paper can improve the efficiency for solving nonlinear eigenvalue problems by the corresponding theoretical analysis and numerical examples.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, Adams (1975)

    MATH  Google Scholar 

  2. Adams, M.F., Bayraktar, H.H., Keaveny, T.M., Papadopoulos, P.: Ultrascalable implicit finite element analyses in solid mechanics with over a half a billion degrees of freedom, In: SC’04: Proceedings of the 2004 ACM/IEEE Conference on Supercomputing, pp. 34–34. IEEE (2004)

  3. Alauzet, F., Frey, P.J.: Estimateur d’erreur géométrique et métriques anisotropes pour l’adaptation de maillage. Partie I: aspects théoriques. Rapport de recherche RR-4759. INRIA, (2003)

  4. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269(5221), 198–201 (1995)

    Article  Google Scholar 

  5. Antoine, X., Levitt, A., Tang, Q.: Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods. J. Comput. Phys. 343, 92–109 (2017)

    Article  MATH  Google Scholar 

  6. Antoine, X., Tang, Q., Zhang, Y.: A preconditioned conjugated gradient method for computing ground states of rotating dipolar Bose–Einstein condensates via kernel truncation method for dipole–dipole interaction evaluation. Commun. Comput. Phys. 24(4), 966–988 (2018)

    Article  MATH  Google Scholar 

  7. Balay, S., Abhyankar, S., Adams, M., Brown, J. et. al.: PETSc users manual revision 3.8, Technical report, Argonne National Lab. (ANL), Argonne, IL (United States) (2017)

  8. Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6(1), 1–135 (2013)

    Article  MATH  Google Scholar 

  9. Bao, W., Du, Q.: Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25(5), 1674–1697 (2004)

    Article  MATH  Google Scholar 

  10. Bao, W., Chern, I.-L., Lim, F.-Y.: Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose–Einstein condensates. J. Comput. Phys. 219(2), 836–854 (2006)

    Article  MATH  Google Scholar 

  11. Bao, G., Hu, G., Liu, D.: Numerical solution of the Kohn–Sham equation by finite element methods with an adaptive mesh redistribution technique. J. Sci. Comput. 55(2), 372–391 (2013)

    Article  MATH  Google Scholar 

  12. Beckett, G., MacKenzie, J., Robertson, M.L.: An \(r\)-adaptive finite element method for the solution of the two-dimensional phase-field equations. Commun. Comput. Phys. 1(5), 805–826 (2006)

    MATH  Google Scholar 

  13. Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer Science & Business Media, Berlin (2007)

    Google Scholar 

  14. Cancès, E., Chakir, R., Maday, Y.: Numerical analysis of nonlinear eigenvalue problems. J. Sci. Comput. 45(1–3), 90–117 (2010)

    Article  MATH  Google Scholar 

  15. Cancès, E., Dusson, G., Maday, Y., Stamm, B., Vohralík, M.: A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations. C. R. Math. 352(11), 941–946 (2014)

    Article  MATH  Google Scholar 

  16. Cancès, E., Chakir, R., He, L., Maday, Y.: Two-grid methods for a class of nonlinear elliptic eigenvalue problems. IMA J. Numer. Anal. 38(2), 605–645 (2018)

    Article  MATH  Google Scholar 

  17. Chen, H.-S., Chang, S.-L., Chien, C.-S.: Spectral collocation methods using sine functions for a rotating Bose–Einstein condensation in optical lattices. J. Comput. Phys. 231(4), 1553–1569 (2012)

    Article  MATH  Google Scholar 

  18. Chien, C.-S., Jeng, B.W.: A two-grid discretization scheme for semilinear elliptic eigenvalue problems. SIAM J. Sci. Comput. 27(4), 1287–1304 (2006)

    Article  MATH  Google Scholar 

  19. Chien, C.-S., Huang, H.-T., Jeng, B.-W., Li, Z.-C.: Two-grid discretization schemes for nonlinear Schrödinger equations. J. Comput. Appl. Math. 214(2), 549–571 (2008)

    Article  MATH  Google Scholar 

  20. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, New Delhi (2002)

    Book  MATH  Google Scholar 

  21. Danaila, I., Kazemi, P.: A new Sobolev gradient method for direct minimization of the Gross–Pitaevskii energy with rotation. SIAM J. Sci. Comput. 32(5), 2447–2467 (2010)

    Article  MATH  Google Scholar 

  22. Danaila, I., Protas, B.: Computation of ground states of the Gross–Pitaevskii functional via Riemannian optimization. SIAM J. Sci. Comput. 39(6), B1102–B1129 (2017)

    Article  MATH  Google Scholar 

  23. Dapogny, C., Dobrzynski, C., Frey, P.: Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems. J. Comput. Phys. 262, 358–378 (2014)

    Article  MATH  Google Scholar 

  24. Davis, K.B., Mewes, M., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M., Ketterle, W.: Bose–Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75(22), 3969 (1995)

    Article  Google Scholar 

  25. Frey, P.-J., Alauzet, F.: Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Eng. 194(48–49), 5068–5082 (2005)

    Article  MATH  Google Scholar 

  26. García-Ripoll, J.J., Pérez-García, V.M.: Optimizing Schrödinger functionals using Sobolev gradients: Applications to quantum mechanics and nonlinear optics. SIAM J. Sci. Comput. 23(4), 1316–1334 (2001)

    Article  MATH  Google Scholar 

  27. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–266 (2012)

    MATH  Google Scholar 

  28. Heid, P., Stamm, B., Wihler, T.P.: Gradient flow finite element discretizations with energy-based adaptivity for the Gross–Pitaevskii equation. J. Comput. Phys. 436, 110165 (2021)

    Article  MATH  Google Scholar 

  29. Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. (TOMS) 31(3), 351–362 (2005)

    Article  MATH  Google Scholar 

  30. Hu, G., Zegeling, P.A.: Simulating finger phenomena in porous media with a moving finite element method. J. Comput. Phys. 230(8), 3249–3263 (2011)

    Article  MATH  Google Scholar 

  31. Hu, G., Qiao, Z., Tang, T.: Moving finite element simulations for reaction-diffusion systems. Adv. Appl. Math. Mech. 4(3), 365–381 (2012)

    Article  Google Scholar 

  32. Jeng, B.W., Chien, C.S., Chern, I.L.: Spectral collocation and a two-level continuation scheme for dipolar Bose–Einstein condensates. J. Comput. Phys. 256, 713–727 (2014)

    Article  MATH  Google Scholar 

  33. Jia, S., Xie, H., Xie, M., Xu, F.: A full multigrid method for nonlinear eigenvalue problems. Sci. China Math. 59(10), 2037–2048 (2016)

    Article  MATH  Google Scholar 

  34. Jolivet, P., Hecht, F., Nataf, F., Prud’Homme, C.: Scalable domain decomposition preconditioners for heterogeneous elliptic problems. Sci. Program. 22(2), 157–171 (2014)

    Google Scholar 

  35. Li, X.-G., Zhu, J., Zhang, R.-P., Cao, S.: A combined discontinuous Galerkin method for the dipolar Bose–Einstein condensation. J. Comput. Phys. 275, 363–376 (2014)

    Article  MATH  Google Scholar 

  36. Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: A rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2000)

    Article  Google Scholar 

  37. Lin, Q., Xie, H.: A multi-level correction scheme for eigenvalue problems. Math. Comput. 84(291), 71–88 (2015)

    Article  MATH  Google Scholar 

  38. Tang, T.: Moving mesh methods for computational fluid dynamics. Contemp. Math. 383(8), 141–173 (2005)

    Article  MATH  Google Scholar 

  39. van Dam, A., Zegeling, P.A.: A robust moving mesh finite volume method applied to 1D hyperbolic conservation laws from magnetohydrodynamics. J. Comput. Phys. 216(2), 526–546 (2006)

    Article  MATH  Google Scholar 

  40. Wang, H., Li, R., Tang, T.: Efficient computation of dendritic growth with \(r\)-adaptive finite element methods. J. Comput. Phys. 227(12), 5984–6000 (2008)

    Article  MATH  Google Scholar 

  41. Wu, X., Wen, Z., Bao, W.: A regularized Newton method for computing ground states of Bose–Einstein condensates. J. Sci. Comput. 73(1), 303–329 (2017)

    Article  MATH  Google Scholar 

  42. Xie, H.: A multigrid method for eigenvalue problem. J. Comput. Phys. 274, 550–561 (2014)

    Article  MATH  Google Scholar 

  43. Xie, H.: A multigrid method for nonlinear eigenvalue problems. Sci Sin. (Mathematica) 45, 1193–1204 (2015)

    Article  MATH  Google Scholar 

  44. Xie, H., Xie, M.: A multigrid method for ground state solution of Bose–Einstein condensates. Commun. Comput. Phys. 19(3), 648–662 (2016)

    Article  MATH  Google Scholar 

  45. Xie, H., Xie, M.: Computable error estimates for ground state solution of Bose–Einstein condensates. J. Sci. Comput. 81(2), 1072–1087 (2019)

    Article  MATH  Google Scholar 

  46. Xu, F.: A cascadic adaptive finite element method for nonlinear eigenvalue problems in quantum physics. Multiscale Model. Simul. 18(1), 198–220 (2020)

    Article  MATH  Google Scholar 

  47. Zhang, N., Xu, F., Xie, H.: An efficient multigrid method for ground state solution of Bose–Einstein condensates. Int. J. Numer. Anal. Model. 16(5), 789–803 (2019)

    MATH  Google Scholar 

  48. Zhou, A.: An analysis of finite-dimensional approximations for the ground state solution of Bose–Einstein condensates. Nonlinearity 17(2), 541–550 (2004)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank both referees for their valuable comments and helpful suggestions that improved this paper.

Funding

The first author (H. Xie) was supported in part by the National Key Research and Development Program of China (2019YFA0709601), Beijing Natural Science Foundation (Z200003) and the National Center for Mathematics and Interdisciplinary Science, CAS. The second author (M. Xie) was supported in part by the National Natural Science Foundation of China (Nos. 12001402, 12071343, 12271400). The third author (X. Yin) was supported by the Hubei Provincial Science and Technology Innovation Base (Platform) Special Project (No. 2020DFH002).

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Authors and Affiliations

  1. LSEC, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Hehu Xie & Gang Zhao

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Hehu Xie & Gang Zhao

  3. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Manting Xie

  4. School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Xiaobo Yin

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  1. Hehu Xie
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Correspondence to Manting Xie.

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Xie, H., Xie, M., Yin, X. et al. An Efficient Adaptive Mesh Redistribution Method for Nonlinear Eigenvalue Problems in Bose–Einstein Condensates. J Sci Comput 94, 37 (2023). https://doi.org/10.1007/s10915-022-02093-2

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  • DOI: https://doi.org/10.1007/s10915-022-02093-2

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