×

zbMATH — the first resource for mathematics

Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials. (English) Zbl 1377.65128
Authors’ abstract: This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The main aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in \(L^\infty(L^2)\) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akrivis, G. D.; Dougalis, V. A.; Karakashian, O. A., On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59, 31-53, (1991) · Zbl 0739.65096
[2] Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184, 2621-2633, (2013) · Zbl 1344.35130
[3] Antoine, X.; Duboscq, R., Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates, J. Comput. Phys., 258, 509-523, (2014) · Zbl 1349.82027
[4] Antoine, X.; Duboscq, R., Gpelab, a Matlab toolbox to solve Gross-Pitaevskii equations. II: dynamics and stochastic simulations, Comput. Phys. Commun., 193, 95-117, (2015) · Zbl 1344.82004
[5] Bao, W.; Cai, Y., Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 50, 492-521, (2012) · Zbl 1246.35188
[6] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1-135, (2013) · Zbl 1266.82009
[7] Bao, W.; Cai, Y., Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comput., 82, 99-128, (2013) · Zbl 1264.65146
[8] Bao, W.; Du, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25, 1674-1697, (2004) · Zbl 1061.82025
[9] Bao, W.; Tang, W., Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187, 230-254, (2003) · Zbl 1028.82500
[10] Cancès, E.; Chakir, R.; Maday, Y., Numerical analysis of nonlinear eigenvalue problems, J. Sci. Comput., 45, 90-117, (2010) · Zbl 1203.65237
[11] Cazenave, T., Semilinear Schrödinger Equations, 10, (2003), Amer. Math. Soc.
[12] Cruz-Uribe, D.; Neugebauer, C. J., Sharp error bounds for the trapezoidal rule and simpson’s rule, JIPAM J. Inequal. Pure Appl. Math., 3, 22, (2002) · Zbl 1030.41016
[13] Danaila, I.; Hecht, F., A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates, J. Comput. Phys., 229, 6946-6960, (2010) · Zbl 1198.82035
[14] Danaila, I.; Kazemi, P., A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32, 2447-2467, (2010) · Zbl 1216.35006
[15] Gauckler, L., Convergence of a split-step Hermite method for the Gross-Pitaevskii equation, IMA J. Numer. Anal., 31, 396-415, (2011) · Zbl 1223.65079
[16] Gilbarg, D.; Trüdinger, N. S., Elliptic Partial Differential Equations of Second Order, (2001), Springer-Verlag · Zbl 1042.35002
[17] Gross, E. P., Structure of a quantized vortex in boson systems, Nuovo Cimento, 20, 454-477, (1961) · Zbl 0100.42403
[18] Henning, P.; Målqvist, A., The finite element method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation, SIAM J. Numer. Anal., 55, 923-952, (2017) · Zbl 1362.65105
[19] Henning, P.; Målqvist, A.; Peterseim, D., Two-level discretization techniques for ground state computations of Bose-Einstein condensates, SIAM J. Numer. Anal., 52, 1525-1550, (2014) · Zbl 1308.35272
[20] Jarlebring, E.; Kvaal, S.; Michiels, W., An inverse iteration method for eigenvalue problems with eigenvector nonlinearities, SIAM J. Sci. Comput., 36, A1978-A2001, (2014) · Zbl 1307.65068
[21] Karakashian, O.; Makridakis, C., A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. Comput., 67, 479-499, (1998) · Zbl 0896.65068
[22] Karakashian, O.; Makridakis, C., A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal., 36, 1779-1807, (1999) · Zbl 0934.65110
[23] Leoni, G., A First Course in Sobolev Spaces, 105, (2009), Amer. Math. Soc. · Zbl 1180.46001
[24] Lieb, E. H.; Seiringer, R.; Yngvason, J., A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Commun. Math. Phys., 224, 17-31, (2001) · Zbl 0996.82010
[25] Lubich, C., On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comput., 77, 2141-2153, (2008) · Zbl 1198.65186
[26] Nikolic, B.; Balaz, A.; Pelster, A., Dipolar Bose-Einstein condensates in weak anisotropic disorder, Phys. Rev. A, 88, 013624, (2013)
[27] Pitaevskii, L. P., Vortex lines in an imperfect Bose gas, Zh. Eksp. Teor. Fiz., 40, 646-651, (1961)
[28] Sanz-Serna, J. M., Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comput., 43, 21-27, (1984) · Zbl 0555.65061
[29] Sataric, B.; Slavnic, V.; Belic, A.; Balaz, A.; Muruganandam, P.; Adhikari, S. K., Hybrid openmp/MPI programs for solving the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun., 200, 411-417, (2016) · Zbl 1351.35004
[30] Thalhammer, M., Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations, SIAM J. Numer. Anal., 50, 3231-3258, (2012) · Zbl 1267.65116
[31] Thalhammer, M.; Abhau, J., A numerical study of adaptive space and time discretisations for Gross-Pitaevskii equations, J. Comput. Phys., 231, 6665-6681, (2012)
[32] Tourigny, Y., Optimal \(H^1\)-estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation, IMA J. Numer. Anal., 11, 509-523, (1991) · Zbl 0737.65095
[33] Vudragovic, D.; Vidanovic, I.; Balaz, A.; Muruganandam, P.; Adhikari, S. K., C programs for solving the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun., 183, 2021-2025, (2012) · Zbl 1353.35003
[34] Wang, J., A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60, 390-407, (2014) · Zbl 1306.65257
[35] Williams, J.; Walser, R.; Wieman, C.; Cooper, J.; Holland, M., Achieving steady-state Bose-Einstein condensation, Phys. Rev. A, 57, 2030-2036, (1998)
[36] Zapata, I.; Sols, F.; Leggett, A. J., Josephson effect between trapped Bose-Einstein condensates, Phys. Rev. A, 57, R28-R31, (1998)
[37] Zouraris, G. E., On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation, M2AN Math. Model. Numer. Anal., 35, 389-405, (2001) · Zbl 0991.65088
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.