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

  • Jianyun Wang (a1) and Yunqing Huang (a2)
Abstract
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 L 2 error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

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*Corresponding author. Email addresses: wjy8137@163.com (J. Y. Wang), huangyq@xtu.edu.cn (Y. Q. Huang)
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[1] AkrivisG. D., Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), pp. 115124.
[2] AkrivisG. D., DougalisV. A. and KarakashianO. A., On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), pp. 3153.
[3] AntoineX., BesseC. and DescombesS., Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 43 (2006), pp. 22722293.
[4] AntoineX., BaoW. Z. and BesseC., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Comm., 184 (2013), pp. 26212633.
[5] AntonopoulouD. C., KaraliG. D., PlexousakisM. and ZourarisG. E., Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain, Math. Comp., 84 (2015), pp. 15711598.
[6] BaoW. Z., JinS. and MarkowichP. A., Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), pp. 2764.
[7] ChangQ. S., JiaE. and SunW., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), pp. 397415.
[8] DehghanM. and TaleeiA., Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method, Numer. Methods Partial Differential Equations, 26 (2010), pp. 979992.
[9] GongX. G., ShenL. H., ZhangD. E. and ZhouA. H., Finite element approximations for Schrödinger equations with applications to electronic structure computations, J. Comput. Math., 26 (2008), pp. 310323.
[10] HuX. L. and ZhangL. M., Conservative compact difference schemes for the coupled nonlinear Schrödinger system, Numer. Methods Partial Differential Equations, 30 (2014), pp. 749772.
[11] IsmailM. S., Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Math. Comput. Simul., 78 (2008), pp. 532547.
[12] JinJ. C., WeiN. and ZhangH. M., A two-grid finite-element method for the nonlinear Schrödinger equation, J. Comput. Math., 33 (2015), pp. 146157.
[13] JinJ. C. and WuX. 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. 777793.
[14] KarakashianO. A., AkrivisG. D. and DougalisV. A., On Optimal Order Error Estimates for the Nonlinear Schrödinger Equation, SIAM J. Numer. Anal., 30 (1993), pp. 377400.
[15] KarakashianO. A. and MakridakisC., A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), pp. 17791807.
[16] KarakashianO. A. and MakridakisC., A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. Comp., 67 (1998), pp. 479499.
[17] LeeH. Y., Fully discrete methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 28 (1994), pp. 924.
[18] LinQ. and LiuX. Q., Global superconvergence estimates of finite element method for Schrödinger equation, J. Comput. Math., 6 (1998), pp. 521526.
[19] LiuY. and LiH., H1-Galerkin mixed finite element method for the linear schrödinger equation, Adv. Math., 39 (2010), pp. 429442.
[20] LUW. Y., HuangY. Q. and LiuH. L., Mass preserving discontinuous Galerkin methods for Schrödinger equations, J. Comput. Phys., 282 (2015), pp. 210226.
[21] SANZ-SernaJ. M., Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comp., 43 (1984), pp. 2127.
[22] ShiD. Y., WangP. L. and ZhaoY. M., Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation, Appl. Math. Lett., 38 (2014), pp. 129134.
[23] TahaT. R., A numerical scheme for the nonlinear Schrödinger equation, Comput. Math. Appl., 22 (1991), pp. 7784.
[24] TourignyY., Optimal H1estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation, IMA J. Numer. Anal., 11 (1991), pp. 509523.
[25] TourignyY. and MorrisJ. 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. 103130.
[26] WangJ. L., A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60 (2014), pp. 390407.
[27] WangJ. Y., HuangY. Q., TianZ. K. and ZhouJ., Superconvergence analysis of finite element method for the time-dependent Schrödinger equation, Comput. Math. Appl., 71 (2016), pp. 19601972.
[28] XuY. and ShuC.W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), pp. 7297.
[29] ZhangH. M., JinJ. C. and WangJ. Y., Two-grid finite-element method for the two-dimensional time-dependent Schrödinger equation, Adv. Appl. Math. Mech., 5 (2013), pp. 180193.
[30] ZhaoY. M., ShiD. Y. and WangF., High accuracy analysis of a new mixed finite element method for nonlinear Schrödinger equation, Math. Numer. Sin., 37 (2015), pp. 162177.
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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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