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Superconvergence analysis of finite element method for time-fractional Thermistor problem

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  • Shi, Dongyang
  • Yang, Huaijun

Abstract

In this paper, the superclose and superconvergence analysis of the nonlinear time-fractional thermistor problem are investigated by bilinear finite element method (FEM) for a fully-discrete scheme, in which the Caputo derivative is approximated by the classical L1 method. By dealing with the error estimates in the spatial direction rigorously, which are one order higher than the traditional FEMs, the superclose estimates in H1-norm are obtained for the corresponding variables based on the special properties of this element together with mean value technique. Subsequently, the global superconvergence results are derived by employing the interpolation postprocessing approach. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

Suggested Citation

  • Shi, Dongyang & Yang, Huaijun, 2018. "Superconvergence analysis of finite element method for time-fractional Thermistor problem," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 31-42.
  • Handle: RePEc:eee:apmaco:v:323:y:2018:i:c:p:31-42
    DOI: 10.1016/j.amc.2017.11.027
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    References listed on IDEAS

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    1. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    2. 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.
    3. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
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    Cited by:

    1. Yin, Baoli & Liu, Yang & Li, Hong, 2020. "A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations," Applied Mathematics and Computation, Elsevier, vol. 368(C).
    2. Shi, Xiangyu & Lu, Linzhang, 2020. "A new two-grid nonconforming mixed finite element method for nonlinear Benjamin-Bona-Mahoney equation," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    3. Gao, Xinghua & Yin, Baoli & Li, Hong & Liu, Yang, 2021. "TT-M FE method for a 2D nonlinear time distributed-order and space fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 117-137.
    4. 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.

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