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Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations

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  • Tan, Zhijun
  • Zeng, Yunhua

Abstract

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2-1σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L2-norm and H1-norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H=h12 and H=hr2r+2, respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods.

Suggested Citation

  • Tan, Zhijun & Zeng, Yunhua, 2024. "Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations," Applied Mathematics and Computation, Elsevier, vol. 466(C).
    • Handle: RePEc:eee:apmaco:v:466:y:2024:i:c:s0096300323006264
    DOI: 10.1016/j.amc.2023.128457
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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. Sun, Hong & Sun, Zhi-zhong & Gao, Guang-hua, 2016. "Some high order difference schemes for the space and time fractional Bloch–Torrey equations," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 356-380.
    3. Wang, Jinfeng & Yin, Baoli & Liu, Yang & Li, Hong & Fang, Zhichao, 2021. "Mixed finite element algorithm for a nonlinear time fractional wave model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 60-76.
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