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Second-order non-uniform and fast two-grid finite element methods for non-linear time-fractional mobile/immobile equations with weak regularity

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  • Tan, Zhijun

Abstract

This paper introduces a novel temporal second-order fully discrete approach of finite element method (FEM) and its fast two-grid FEM on non-uniform meshes, which aims to solve non-linear time-fractional variable coefficient mobile/immobile (MIM) equations with a solution exhibiting weak regularity. The proposed method utilizes the averaged L1 formula on graded meshes in the temporal domain to handle the weak initial singularity. In the spatial domain, a two-grid approach based on FEM and its associated fast algorithm are employed to optimize computational efficiency. To ensure fast and accurate calculations of kernels, an innovative algorithm is developed. The stability and optimal error estimates in L2-norm and H1-norm are rigorously established for the non-uniform averaged L1-based FEM, two-grid FEM and their associated fast algorithms, respectively. The numerical findings clearly showcase the validity of our theoretical discoveries, highlighting the enhanced effectiveness of our two-grid approach in contrast to the conventional approach. An important point to mention is that this work is the pioneering effort in addressing both H1-stability and second-order H1-norm error analysis for the fractional MIM problem with weak regularity, as well as temporal second-order approaches of two-grid for the fractional MIM equation with a weakly singular solution.

Suggested Citation

  • Tan, Zhijun, 2025. "Second-order non-uniform and fast two-grid finite element methods for non-linear time-fractional mobile/immobile equations with weak regularity," Applied Mathematics and Computation, Elsevier, vol. 486(C).
    • Handle: RePEc:eee:apmaco:v:486:y:2025:i:c:s0096300324005046
    DOI: 10.1016/j.amc.2024.129043
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    References listed on IDEAS

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    1. She, Mianfu & Li, Dongfang & Sun, Hai-wei, 2022. "A transformed L1 method for solving the multi-term time-fractional diffusion problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 584-606.
    2. 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).
    3. Zeng, Yunhua & Tan, Zhijun, 2022. "Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    4. 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.
    5. Li, Qingfeng & Chen, Yanping & Huang, Yunqing & Wang, Yang, 2021. "Two-grid methods for nonlinear time fractional diffusion equations by L1-Galerkin FEM," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 436-451.
    6. Zhang, Jiali & Fang, Zhi-Wei & Sun, Hai-Wei, 2022. "Robust fast method for variable-order time-fractional diffusion equations without regularity assumptions of the true solutions," Applied Mathematics and Computation, Elsevier, vol. 430(C).
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    More about this item

    Keywords

    Two-grid FEM; Non-linear time-fractional MIM equations; Non-uniform averaged L1 formula; Fast algorithm; Error analysis;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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