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A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator

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  • Li, Meng
  • Zhao, Yong-Liang

Abstract

The main aim of this paper is to apply the Galerkin finite element method to numerically solve the nonlinear fractional Schrödinger equation with wave operator. We first construct a fully discrete scheme combining the Crank–Nicolson method with the Galerkin finite element method. Two conserved quantities of the discrete system are shown. Meanwhile, the prior bound of the discrete solutions are proved. Then, we prove that the discrete scheme is unconditionally convergent in the senses of L2−norm and Hα/2−norm. Moreover, by the proposed iterative algorithm, some numerical examples are given to verify the theoretical results and show the effectiveness of the numerical scheme. Finally, a fast Krylov subspace solver with suitable circulant preconditioner is designed to solve above Toeplitz-like linear system. In each iterative step, this method can effectively reduce the memory requirement of the proposed iterative finite element scheme from O(M2) to O(M), and the computational complexity from O(M3) to O(MlogM), where M is the number of grid nodes. Several numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization methods, in terms of memory requirement and computational cost.

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  • Li, Meng & Zhao, Yong-Liang, 2018. "A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 758-773.
    • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:758-773
    DOI: 10.1016/j.amc.2018.06.010
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    References listed on IDEAS

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    1. Wang, Dongling & Xiao, Aiguo & Yang, Wei, 2015. "Maximum-norm error analysis of a difference scheme for the space fractional CNLS," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 241-251.
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    Cited by:

    1. He, Tingxiao & Wang, Yun & Zhang, Yingnan, 2024. "A partial-integrable numerical simulation scheme of the derivative nonlinear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 630-639.
    2. Li, Meng & Fei, Mingfa & Wang, Nan & Huang, Chengming, 2020. "A dissipation-preserving finite element method for nonlinear fractional wave equations on irregular convex domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 404-419.
    3. Bzeih, Moussa & Arwadi, Toufic El & Wehbe, Ali & Madureira, Rodrigo L.R. & Rincon, Mauro A., 2023. "A finite element scheme for a 2D-wave equation with dynamical boundary control," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 315-339.
    4. Li, Meng & Wei, Yifan & Niu, Binqian & Zhao, Yong-Liang, 2022. "Fast L2-1σ Galerkin FEMs for generalized nonlinear coupled Schrödinger equations with Caputo derivatives," Applied Mathematics and Computation, Elsevier, vol. 416(C).
    5. Fu, Yayun & Song, Yongzhong & Wang, Yushun, 2019. "Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 206-223.
    6. Liu, Yang & Ran, Maohua, 2024. "Arbitrarily high-order explicit energy-conserving methods for the generalized nonlinear fractional Schrödinger wave equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 126-144.
    7. You, Xiangcheng & Xu, Hang & Sun, Qiang, 2022. "Analysis of BBM solitary wave interactions using the conserved quantities," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    8. Zou, Guang-an & Wang, Bo & Sheu, Tony W.H., 2020. "On a conservative Fourier spectral Galerkin method for cubic nonlinear Schrödinger equation with fractional Laplacian," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 168(C), pages 122-134.
    9. Almushaira, Mustafa, 2023. "An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 424-447.

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