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

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  • 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
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    References listed on IDEAS

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    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. Iomin, Alexander, 2011. "Fractional-time Schrödinger equation: Fractional dynamics on a comb," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 348-352.
    4. 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.
    5. 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.
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