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Unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation

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  • Shi, Dongyang
  • Wang, Junjun

Abstract

Galerkin finite element approximation to nonlinear parabolic equation is studied with a linearized backward Euler scheme. The error between the exact solution and the numerical solution is split into two parts which are called the temporal error and the spatial error through building a time-discrete system. On one hand, the temporal error derived skillfully leads to the regularity of the time-discrete system solution. On the other hand, the τ-independent spatial error and the boundedness of the numerical solution in L∞-norm is deduced with the above achievements. At last, the superclose result of order O(h2+τ) in H1-norm is obtained without any restriction of τ in a routine way. Here, h is the subdivision parameter, and τ, the time step.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:294:y:2017:i:c:p:216-226
    DOI: 10.1016/j.amc.2016.08.024
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    References listed on IDEAS

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    1. 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.
    2. Shi, Dongyang & Tang, Qili & Gong, Wei, 2015. "A low order characteristic-nonconforming finite element method for nonlinear Sobolev equation with convection-dominated term," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 114(C), pages 25-36.
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    Citations

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    Cited by:

    1. 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.
    2. 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).
    3. 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).
    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. 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.
    6. 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).
    7. 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).

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