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A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

Author

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  • Jie Zhao

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
    School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China)

  • Hong Li

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China)

  • Zhichao Fang

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China)

  • Yang Liu

    (School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China)

Abstract

In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

Suggested Citation

  • Jie Zhao & Hong Li & Zhichao Fang & Yang Liu, 2019. "A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids," Mathematics, MDPI, vol. 7(7), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:600-:d:246062
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    References listed on IDEAS

    as
    1. Zhao, Yanmin & Bu, Weiping & Huang, Jianfei & Liu, Da-Yan & Tang, Yifa, 2015. "Finite element method for two-dimensional space-fractional advection–dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 553-565.
    2. Zhang, Tie & Li, Zheng, 2015. "An analysis of finite volume element method for solving the Signorini problem," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 830-841.
    3. Metzler, Ralf & Klafter, Joseph, 2000. "Boundary value problems for fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(1), pages 107-125.
    4. Feng, L.B. & Zhuang, P. & Liu, F. & Turner, I., 2015. "Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 52-65.
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    Cited by:

    1. Jie Zhao & Zhichao Fang & Hong Li & Yang Liu, 2020. "A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids," Mathematics, MDPI, vol. 8(9), pages 1-17, September.

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