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Volume 40, Issue 6
A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations

Yanping Chen, Qiling Gu, Qingfeng Li & Yunqing Huang

DOI: 10.4208/jcm.2104-m2021-0332

J. Comp. Math., 40 (2022), pp. 936-954.

Published online: 2022-08

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  • Abstract

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 $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-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.

  • Keywords

Two-grid method, Finite element method, Nonlinear time fractional mixed sub-diffusion and diffusion-wave equations, L1-CN scheme, Stability and convergence.

  • AMS Subject Headings

65N30, 65M60, 26A33

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yanpingchen@scnu.edu.cn (Yanping Chen)

962601731@qq.com (Qiling Gu)

lqfmath@163.com (Qingfeng Li)

huangyq@xtu.edu.cn (Yunqing Huang)

  • BibTex
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  • TXT
@Article{JCM-40-936, author = {Chen , YanpingGu , QilingLi , Qingfeng and Huang , Yunqing}, title = {A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {6}, pages = {936--954}, abstract = {

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 $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2104-m2021-0332}, url = {http://global-sci.org/intro/article_detail/jcm/20842.html} }
TY - JOUR T1 - A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations AU - Chen , Yanping AU - Gu , Qiling AU - Li , Qingfeng AU - Huang , Yunqing JO - Journal of Computational Mathematics VL - 6 SP - 936 EP - 954 PY - 2022 DA - 2022/08 SN - 40 DO - http://doi.org/10.4208/jcm.2104-m2021-0332 UR - https://global-sci.org/intro/article_detail/jcm/20842.html KW - Two-grid method, Finite element method, Nonlinear time fractional mixed sub-diffusion and diffusion-wave equations, L1-CN scheme, Stability and convergence. AB -

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 $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-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.

Chen , YanpingGu , QilingLi , Qingfeng and Huang , Yunqing. (2022). A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations. Journal of Computational Mathematics. 40 (6). 936-954. doi:10.4208/jcm.2104-m2021-0332
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