IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v416y2022ics009630032100816x.html
   My bibliography  Save this article

Fast L2-1σ Galerkin FEMs for generalized nonlinear coupled Schrödinger equations with Caputo derivatives

Author

Listed:
  • Li, Meng
  • Wei, Yifan
  • Niu, Binqian
  • Zhao, Yong-Liang

Abstract

The paper is concerned with the unconditional stability and optimal error estimates of Galerkin finite element methods (FEMs) for a class of generalized nonlinear coupled Schrödinger equations with Caputo-type derivatives. We improve the results in [1] to a higher order temporal scheme by using a type of new Grönwall inequality. By introducing a time-discrete system, the error is separated into two parts: the temporal error and the spatial error. As the result of τ-independent of the spatial error, we obtain the L∞-norm boundedness of the fully discrete solutions without any restrictions on the grid ratio. The unconditionally optimal L2-norm error estimate is then obtained naturally. Furthermore, in order to numerically solve the system with nonsmooth solutions, we construct another Galerkin FEM with nonuniform temporal meshes, and corresponding fast algorithm by using sum-of-exponentials technique is also built. Finally, numerical results are reported to show the accuracy and efficiency of the proposed FEMs and the corresponding fast algorithms.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:416:y:2022:i:c:s009630032100816x
    DOI: 10.1016/j.amc.2021.126734
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S009630032100816X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2021.126734?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Yimin Zhao, 2017. "Space as method," City, Taylor & Francis Journals, vol. 21(2), pages 190-206, March.
    3. 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.
    4. 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.
    5. Iomin, Alexander, 2011. "Fractional-time Schrödinger equation: Fractional dynamics on a comb," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 348-352.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xu, Chao & Shi, Dongyang, 2019. "Superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 1-11.
    2. Shi, Dongyang & Yang, Huaijun, 2019. "Superconvergence analysis of nonconforming FEM for nonlinear time-dependent thermistor problem," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 210-224.
    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. Wu, Yanmi & Shi, Dongyang, 2021. "Quasi-uniform and unconditional superconvergence analysis of Ciarlet–Raviart scheme for the fourth order singularly perturbed Bi-wave problem modeling d-wave superconductors," Applied Mathematics and Computation, Elsevier, vol. 397(C).
    5. Yaxin Hou & Cao Wen & Hong Li & Yang Liu & Zhichao Fang & Yining Yang, 2020. "Some Second-Order σ Schemes Combined with an H 1 -Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation," Mathematics, MDPI, vol. 8(2), pages 1-19, February.
    6. Zhao, Zhihui & Li, Hong & Wang, Jing, 2021. "The study of a continuous Galerkin method for Sobolev equation with space-time variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    7. 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.
    8. Li, Zhenzhen & Li, Minghao & Shi, Dongyang, 2021. "Unconditional convergence and superconvergence analysis for the transient Stokes equations with damping," Applied Mathematics and Computation, Elsevier, vol. 389(C).
    9. Shi, Dongyang & Yang, Huaijun, 2017. "Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 40-47.
    10. Hao, Zhaopeng & Lin, Guang & Zhang, Zhongqiang, 2020. "Error estimates of a spectral Petrov–Galerkin method for two-sided fractional reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 374(C).
    11. Iomin, Alexander, 2023. "Fractional Floquet theory," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    12. Gong, Yujie & Yuan, Guangwei & Cui, Xia, 2024. "Analysis on a high accuracy fully implicit solution for strong nonlinear diffusion problem - convergence, stability, and uniqueness," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    13. Kai Liu & Yu Liang & Hong S. He & Wen J. Wang & Chao Huang & Shengwei Zong & Lei Wang & Jiangtao Xiao & Haibo Du, 2018. "Long-Term Impacts of China’s New Commercial Harvest Exclusion Policy on Ecosystem Services and Biodiversity in the Temperate Forests of Northeast China," Sustainability, MDPI, vol. 10(4), pages 1-16, April.
    14. Seongwoo Jeon & Hyunjung Hong & Sungdae Kang, 2018. "Simulation of Urban Growth and Urban Living Environment with Release of the Green Belt," Sustainability, MDPI, vol. 10(9), pages 1-21, September.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. You, Xiangcheng & Xu, Hang & Sun, Qiang, 2022. "Analysis of BBM solitary wave interactions using the conserved quantities," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    20. Li, Lili & Zhao, Dan & She, Mianfu & Chen, Xiaoli, 2022. "On high order numerical schemes for fractional differential equations by block-by-block approach," Applied Mathematics and Computation, Elsevier, vol. 425(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:416:y:2022:i:c:s009630032100816x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.