Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T07:03:55.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  20 June 2017

Jianyun Wang  and
Yunqing Huang
Jianyun Wang*
Affiliation:
School of Science, Hunan University of Technology, Zhuzhou 412007, China
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
*
*Corresponding author. Email addresses:[email protected] (J. Y. Wang), [email protected] (Y. Q. Huang)
*Corresponding author. Email addresses:[email protected] (J. Y. Wang), [email protected] (Y. Q. Huang)
Get access

Abstract

This paper is concerned with numerical method for a two-dimensional time-dependent cubic nonlinear Schrödinger equation. The approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time, respectively. We prove optimal L2 error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

Keywords

Finite element methodnonlinear Schrödinger equationsbackward Euler schemeCrank-Nicolson scheme

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M50: Mesh generation and refinement 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 671 - 688
Copyright
Copyright © Global-Science Press 2017 

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] Akrivis, G. D., Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), pp. 115–124.CrossRefGoogle Scholar
[2] Akrivis, G. D., Dougalis, V. A. and Karakashian, O. A., On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), pp. 31–53.Google Scholar
[3] Antoine, X., Besse, C. and Descombes, S., Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 43 (2006), pp. 2272–2293.Google Scholar
[4] Antoine, X., Bao, W. Z. and Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Comm., 184 (2013), pp. 2621–2633.Google Scholar
[5] Antonopoulou, D. C., Karali, G. D., Plexousakis, M. and Zouraris, G. E., Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain, Math. Comp., 84 (2015), pp. 1571–1598.Google Scholar
[6] Bao, W. Z., Jin, S. and Markowich, P. A., Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), pp. 27–64.CrossRefGoogle Scholar
[7] Chang, Q. S., Jia, E. and Sun, W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), pp. 397–415.Google Scholar
[8] Dehghan, M. and Taleei, A., Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method, Numer. Methods Partial Differential Equations, 26 (2010), pp. 979–992.CrossRefGoogle Scholar
[9] Gong, X. G., Shen, L. H., Zhang, D. E. and Zhou, A. H., Finite element approximations for Schrödinger equations with applications to electronic structure computations, J. Comput. Math., 26 (2008), pp. 310–323.Google Scholar
[10] Hu, X. L. and Zhang, L. M., Conservative compact difference schemes for the coupled nonlinear Schrödinger system, Numer. Methods Partial Differential Equations, 30 (2014), pp. 749–772.Google Scholar
[11] Ismail, M. S., Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Math. Comput. Simul., 78 (2008), pp. 532–547.Google Scholar
[12] Jin, J. C., Wei, N. and Zhang, H. M., A two-grid finite-element method for the nonlinear Schrödinger equation, J. Comput. Math., 33 (2015), pp. 146–157.Google Scholar
[13] Jin, J. C. and Wu, X. N., Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip, J. Comput. Appl. Math., 234 (2010), pp. 777–793.Google Scholar
[14] Karakashian, O. A., Akrivis, G. D. and Dougalis, V. A., On Optimal Order Error Estimates for the Nonlinear Schrödinger Equation, SIAM J. Numer. Anal., 30 (1993), pp. 377–400.CrossRefGoogle Scholar
[15] Karakashian, O. A. and Makridakis, C., A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), pp. 1779–1807.CrossRefGoogle Scholar
[16] Karakashian, O. A. and Makridakis, C., A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. Comp., 67 (1998), pp. 479–499.Google Scholar
[17] Lee, H. Y., Fully discrete methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 28 (1994), pp. 9–24.Google Scholar
[18] Lin, Q. and Liu, X. Q., Global superconvergence estimates of finite element method for Schrödinger equation, J. Comput. Math., 6 (1998), pp. 521–526.Google Scholar
[19] Liu, Y. and Li, H., H1-Galerkin mixed finite element method for the linear schrödinger equation, Adv. Math., 39 (2010), pp. 429–442.Google Scholar
[20] LU, W. Y., Huang, Y. Q. and Liu, H. L., Mass preserving discontinuous Galerkin methods for Schrödinger equations, J. Comput. Phys., 282 (2015), pp. 210–226.Google Scholar
[21] SANZ-Serna, J. M., Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comp., 43 (1984), pp. 21–27.CrossRefGoogle Scholar
[22] Shi, D. Y., Wang, P. L. and Zhao, Y. M., Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation, Appl. Math. Lett., 38 (2014), pp. 129–134.Google Scholar
[23] Taha, T. R., A numerical scheme for the nonlinear Schrödinger equation, Comput. Math. Appl., 22 (1991), pp. 77–84.Google Scholar
[24] Tourigny, Y., Optimal H1estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation, IMA J. Numer. Anal., 11 (1991), pp. 509–523.CrossRefGoogle Scholar
[25] Tourigny, Y. and Morris, J. L., An investigation into the effect of product approximation in the numerical solution of the cubic nonlinear Schrödinger equation, J. Comput. Phys., 76 (1988), pp. 103–130.Google Scholar
[26] Wang, J. L., A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60 (2014), pp. 390–407.Google Scholar
[27] Wang, J. Y., Huang, Y. Q., Tian, Z. K. and Zhou, J., Superconvergence analysis of finite element method for the time-dependent Schrödinger equation, Comput. Math. Appl., 71 (2016), pp. 1960–1972.CrossRefGoogle Scholar
[28] Xu, Y. and Shu, C.W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), pp. 72–97.Google Scholar
[29] Zhang, H. M., Jin, J. C. and Wang, J. Y., Two-grid finite-element method for the two-dimensional time-dependent Schrödinger equation, Adv. Appl. Math. Mech., 5 (2013), pp. 180–193.Google Scholar
[30] Zhao, Y. M., Shi, D. Y. and Wang, F., High accuracy analysis of a new mixed finite element method for nonlinear Schrödinger equation, Math. Numer. Sin., 37 (2015), pp. 162–177.Google Scholar