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Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation

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  • Shi, Dongyang
  • Liao, Xin
  • Wang, Lele

Abstract

The main aim of this paper is to apply the conforming bilinear finite element to solve the nonlinear Schrödinger equation (NLSE). Firstly, the stability and convergence for time discrete scheme are proved. Secondly, through a new estimate approach, the optimal order error estimates and superclose properties in H1-norm are obtained with Backward Euler (B-E) and Crank-Nicolson (C-N) fully-discrete schemes, the global superconvergence results are deduced with the help of interpolation postprocessing technique. Finally, some numerical examples are provided to verify the theoretical analysis.

Suggested Citation

  • Shi, Dongyang & Liao, Xin & Wang, Lele, 2016. "Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 298-310.
    • Handle: RePEc:eee:apmaco:v:289:y:2016:i:c:p:298-310
    DOI: 10.1016/j.amc.2016.05.020
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    References listed on IDEAS

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    1. Dehghan, Mehdi, 2006. "Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 71(1), pages 16-30.
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    Cited by:

    1. Shi, Xiangyu & Lu, Linzhang, 2020. "A new two-grid nonconforming mixed finite element method for nonlinear Benjamin-Bona-Mahoney equation," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    2. Shi, Dongyang & Wang, Junjun, 2017. "Unconditional superconvergence analysis for nonlinear hyperbolic equation with nonconforming finite element," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 1-16.
    3. Zhang, Houchao & Shi, Dongyang & Li, Qingfu, 2020. "Nonconforming finite element method for a generalized nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    4. Shi, Dongyang & Yang, Huaijun, 2018. "Superconvergence analysis of finite element method for time-fractional Thermistor problem," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 31-42.
    5. Fang, Yin & Bo, Wen-Bo & Wang, Ru-Ru & Wang, Yue-Yue & Dai, Chao-Qing, 2022. "Predicting nonlinear dynamics of optical solitons in optical fiber via the SCPINN," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    6. Shi, Dongyang & Wang, Junjun, 2017. "Unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 216-226.

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