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立方Schrödinger方程的半隐格式BDF2-FEM无条件最优误差估计

代猛 尹小艳

代猛, 尹小艳. 立方Schrödinger方程的半隐格式BDF2-FEM无条件最优误差估计[J]. 应用数学和力学, 2019, 40(6): 663-681. doi: 10.21656/1000-0887.390209
引用本文: 代猛, 尹小艳. 立方Schrödinger方程的半隐格式BDF2-FEM无条件最优误差估计[J]. 应用数学和力学, 2019, 40(6): 663-681. doi: 10.21656/1000-0887.390209
DAI Meng, YIN Xiaoyan. Unconditionally Optimal Error Estimates of the Semi-Implicit BDF2-FEM for Cubic Schrödinger Equations[J]. Applied Mathematics and Mechanics, 2019, 40(6): 663-681. doi: 10.21656/1000-0887.390209
Citation: DAI Meng, YIN Xiaoyan. Unconditionally Optimal Error Estimates of the Semi-Implicit BDF2-FEM for Cubic Schrödinger Equations[J]. Applied Mathematics and Mechanics, 2019, 40(6): 663-681. doi: 10.21656/1000-0887.390209

立方Schrödinger方程的半隐格式BDF2-FEM无条件最优误差估计

doi: 10.21656/1000-0887.390209
基金项目: 国家自然科学基金(面上项目)(11771259);中央高校基础科研业务费(JB180714)
详细信息
    作者简介:

    代猛(1993—), 男, 硕士生(E-mail: dm1614720343@163.com);尹小艳(1979—), 女, 副教授, 博士, 硕士生导师(通讯作者. E-mail: xyyin@mail.xidian.edu.cn).

  • 中图分类号: O241.82

Unconditionally Optimal Error Estimates of the Semi-Implicit BDF2-FEM for Cubic Schrödinger Equations

Funds: The National Natural Science Foundation of China(General Program)(11771259)
  • 摘要: 研究了立方Schrödinger方程的二阶向后差分有限元方法(BDF2-FEM)的无条件最优误差估计.首先,将误差分为时间误差和空间误差两部分.通过引入时间离散方程,得到时间离散方程解的一致有界性,并给出时间误差估计.从而得到该方程在半隐格式下BDF2FEM无条件最优误差估计.最后,用数值算例验证了理论分析.
  • [1] DELFOUR M, FORTIN M, PAYRE G. Finite-difference solutions of a non-linear Schrdinger equation[J]. Journal of Computational Physics,1981,44(2): 277-288.
    [2] EBAID A, KHALED S M. New types of exact solutions for nonlinear Schrdinger equation with cubic nonlinearity[J]. Journal of Computational and Applied Mathematics,2011,235(8): 1984-1992.
    [3] LI B Y, SUN W W. Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations[J]. International Journal of Numerical Analysis and Modeling,2013,10(3): 622-633.
    [4] LI B Y, SUN W W. Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations[J]. SIAM Journal on Numerical Analysis,2012,52(2): 933-954.
    [5] LAMBERT J D. Numerical Methods for Ordinary Differential Systems: the Initial Value Problem [J]. New York: John Wiley & Sons Inc, 1991.
    [6] BAKER G, DOUGALIS V, KARAKASHIAN O. On a higher accurate fully discrete Galerkin approximation to the Navier-Stokes equations[J]. Mathematics of Computation,1982,39(160): 339-375.
    [7] CAI W, LI J, CHEN Z. Unconditional convergence and optimal error estimates of the Euler semi-implicit scheme for a generalized nonlinear Schrdinger equation[J]. Advances in Computational Mathematics,2016,42(6): 1311-1330.
    [8] CAI W, LI J, CHEN Z. Unconditional optimal error estimates for BDF2-FEM for a nonlinear Schrdinger equation[J]. Journal of Computational and Applied Mathematics,2018,331: 23-41.
    [9] DUPONT T. Three-level Galerkin methods for parabolic equations[J]. SIAM Journal on Numerical Analysis,1974,11(2): 392-410.
    [10] 姜礼尚, 庞之垣. 有限元方法及其理论[M]. 北京: 人民教育出版社, 1979.(JIANG Lishang, PANG Zhiyuan. Finite Element Method and Its Theory [M]. Beijing: People’s Education Press, 1979.(in Chinese))
    [11] BREZZI F, RAPPAZ J, RAVIART P A. Finite Dimensional Approximation of Nonlinear Problems [M]. New York: Springer-Verlag, 1980.
    [12] AKRIVIS G, LARSSON S. Linearly implicit finite element methods for the time-dependent Joule heating problem[J]. Bit Numerical Mathematics,2005,45(3): 429-442.
    [13] JENSEN M, MALQVIST A. Finite element convergence for the Joule heating problem with mixed boundary conditions[J]. Bit Numerical Mathematics,2013, 53(2): 475-496.
    [14] BULUT H, PANDIR Y, DEMIRAY S T. Exact solutions of nonlinear Schrdinger equation with dual power-law nonlinearity by extended trial equation method[J]. Waves Random Complex Media,2014,24(4): 439-451.
    [15] HEYWOOD J G, RANNACHER R. Finite element approximation of the nonstationary Navier-Stokes problem IV: error analysis for second-order time discretization[J]. SIAM Journal on Numerical Analysis,1984,27(2): 353-384.
    [16] FEIT M D, FLECK J A, STEIGER A. Solution of the Schrdinger equation by a spectral method II: vibrational energy levels of triatomic molecules[J]. Journal of Computational Physics,1983,78(1): 301-308.
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出版历程
  • 收稿日期:  2018-07-31
  • 修回日期:  2019-04-13
  • 刊出日期:  2019-06-01

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