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 - <p style="text-align: justify;">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.</p>