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A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

Yanping Chen1, Qiling Gu2, Qingfeng Li2, Yunqing Huang2   

  1. 1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
    2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411199, China
  • Received:2020-12-25 Revised:2021-01-30 Online:2022-11-15 Published:2022-11-18
  • Contact: Yanping Chen, Email: yanpingchen@scnu.edu.cn
  • Supported by:
    This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133, 11971410), Project for Hunan National Applied Mathematics Center of Hunan Provincial Science and Technology Department (2020ZYT003), Hunan Provincial Innovation Foundation for Postgraduate, China (XDCX2020B082, XDCX2021B098), Postgraduate Scientific Research Innovation Project of Hunan Province (CX20210597).

Yanping Chen, Qiling Gu, Qingfeng Li, Yunqing Huang. A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS[J]. Journal of Computational Mathematics, 2022, 40(6): 936-954.

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order α ∈ (1, 2) and α1 ∈ (0, 1). Numerical stability and optimal error estimate O(hr+1 + H2r+2 + τmin-3-α,2-α1}) in L2-norm are presented for two-grid scheme, where t, H and h are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

CLC Number: 

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