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An α-robust two-grid finite element method with nonuniform L2-1σ scheme for the semilinear Caputo-Hadamard time-fractional diffusion equations involving initial singularity

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  • Zeng, Yunhua
  • Tan, Zhijun

Abstract

Considering the initial singularity, a fully discrete two-grid finite element method (FEM) on nonuniform temporal meshes is constructed for the semilinear time-fractional variable coefficient diffusion equations (TF-VCDEs) with Caputo-Hadamard derivative. The nonuniform Llog⁡,2−1σ formula and two-grid method are employed to discretize the time and space directions, respectively. Through strict theoretical proof, the α-robust stability and optimal L2-norm and H1-norm error analysis for the fully discrete FEM and the two-grid method are obtained, where the error bound does not blow up as α→1−. To reduce computational costs, a fast two-grid method is constructed by approximating the kernel function with an effective sum-of-exponentials (SOE) technique. Finally, the accuracy and effectiveness of the two-grid method and its associated fast algorithm are verified through two numerical examples.

Suggested Citation

  • Zeng, Yunhua & Tan, Zhijun, 2025. "An α-robust two-grid finite element method with nonuniform L2-1σ scheme for the semilinear Caputo-Hadamard time-fractional diffusion equations involving initial singularity," Applied Mathematics and Computation, Elsevier, vol. 496(C).
    • Handle: RePEc:eee:apmaco:v:496:y:2025:i:c:s0096300325000827
    DOI: 10.1016/j.amc.2025.129355
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    References listed on IDEAS

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    1. Li, Binjie & Xie, Xiaoping & Zhang, Shiquan, 2019. "A new smoothness result for Caputo-type fractional ordinary differential equations," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 408-420.
    2. Li, Meng & Wei, Yifan & Niu, Binqian & Zhao, Yong-Liang, 2022. "Fast L2-1σ Galerkin FEMs for generalized nonlinear coupled Schrödinger equations with Caputo derivatives," Applied Mathematics and Computation, Elsevier, vol. 416(C).
    3. J. A. Tenreiro Machado & Manuel F. Silva & Ramiro S. Barbosa & Isabel S. Jesus & Cecília M. Reis & Maria G. Marcos & Alexandra F. Galhano, 2010. "Some Applications of Fractional Calculus in Engineering," Mathematical Problems in Engineering, Hindawi, vol. 2010, pages 1-34, November.
    4. Zhang, Haiqing & Liao, Hong-lin, 2024. "High-order energy stable variable-step schemes for the time-fractional Cahn–Hilliard model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 171-182.
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