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Unconditional superconvergence analysis for nonlinear hyperbolic equation with nonconforming finite element

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  • Shi, Dongyang
  • Wang, Junjun

Abstract

Nonlinear hyperbolic equation is studied by developing a linearized Galerkin finite element method (FEM) with nonconforming EQ1rot element. A time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully which leads to the analysis for the regularity of the time-discrete system. The spatial error is derived τ-independently with order O(h2+hτ) in broken H1-norm. The final unconditional superclose result of u with order O(h2+τ2) is deduced based on the above achievements. The two typical characters of this nonconforming EQ1rot element (see Lemma 1 below) play an important role in the procedure of proof. At last, a numerical example is provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ, the time step.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:305:y:2017:i:c:p:1-16
    DOI: 10.1016/j.amc.2017.01.050
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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. 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.
    3. 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.
    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. 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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