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Superconvergence Analysis of Anisotropic FEMs for Time Fractional Variable Coefficient Diffusion Equations

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Abstract

In this paper, based on Alikhanov’s L2-\(1_\sigma \) high-order approximation and anisotropic finite element methods, a fully discrete scheme for time fractional variable coefficient diffusion equations on anisotropic meshes is presented. Firstly, we prove that the discrete scheme is unconditionally stable in \(H^1\)-norm, then the results of convergence in \(L_2\)-norm and superclose in \(H^1\)-norm are derived by combining interpolation with projection, and then, the superconvergence in \(H^1\)-norm is obtained by using interpolation post-processing technique. In addition, it is worth mentioning the key technology of combining interpolation and projection. If interpolation or projection is used alone, the results of this article cannot be obtained. Finally, numerical examples are provided to verify the correctness of theoretical analysis.

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Acknowledgements

The work is supported by the National Natural Science Foundation of China (Nos. 11971416, 11771438), the Specialized Research Fund for State Key Laboratory of Space Weather (No. 201916), the Key Scientific Research Projects in Universities of Henan Province (No. 19B110013), the Program for Scientific and Technological Innovation Talents in Universities of Henan Province (No. 19HASTIT025) and Natural Science Foundation of Ningxia (No. NZ15041). We express our gratitude to Prof. Fawang Liu for his valuable suggestion.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beihang University, Beijing, 100083, China

    Yabing Wei

  2. State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing, 100190, China

    Yabing Wei

  3. School of Mathematics and Statistics, Xuchang University, Xuchang, 461000, China

    Yanmin Zhao & Fenling Wang

  4. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yifa Tang

  5. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yifa Tang

  6. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

    Jiye Yang

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  1. Yabing Wei
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  2. Yanmin Zhao
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  3. Fenling Wang
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Correspondence to Yanmin Zhao.

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Communicated by Theodore E. Simos.

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Wei, Y., Zhao, Y., Wang, F. et al. Superconvergence Analysis of Anisotropic FEMs for Time Fractional Variable Coefficient Diffusion Equations. Bull. Malays. Math. Sci. Soc. 43, 4411–4429 (2020). https://doi.org/10.1007/s40840-020-00929-4

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  • DOI: https://doi.org/10.1007/s40840-020-00929-4

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