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Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations

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Abstract

Based on spatial finite-element methods combined with classical L1 time stepping method, the superconvergence analysis of the two-grid approximate scheme for the two-dimensional nonlinear time-fractional diffusion equations is considered. First, we use the rectangular Lagrange type finite element of order p to get a two-grid fully discrete scheme of the equation and discuss the superclose error estimate with the order \(O\left( h^{p+1}+H^{2p+2}+\tau ^{2-\alpha }\right) \) in the \(H^{1}\) norm, here \(\tau \), H and h denote time step, coarse and fine grid sizes, respectively. Second, through the interpolated postprocessing approach, the global superconvergence of order \(O\left( h^{2}+H^{4}+\tau ^{2-\alpha }\right) \) in the \(H^{1}\) norm is obtained. Finally, two numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Yanping Chen

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Qiling Gu & Yunqing Huang

Authors
  1. Qiling Gu
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  2. Yanping Chen
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  3. Yunqing Huang
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Correspondence to Yanping Chen.

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Communicated by Frederic Valentin.

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This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133, 11971410, 11671157), Project for Hunan National Applied Mathematics Center of Hunan Provincial Science and Technology Department (2020ZYT003), Hunan Provincial Innovation Foundation for Postgraduate, China (XDCX2021B098), Postgraduate Scientific Research Innovation Project of Hunan Province (CX20210597)

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Gu, Q., Chen, Y. & Huang, Y. Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations. Comp. Appl. Math. 41, 361 (2022). https://doi.org/10.1007/s40314-022-02070-3

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  • DOI: https://doi.org/10.1007/s40314-022-02070-3

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