Variable-step L1 method combined with time two-grid algorithm for multi-singularity problems arising from two-dimensional nonlinear delay fractional equations

Abstract

In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at ${\tau }^{+}$ is smoother than that at $0^{+}$, where $\tau$ is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on $L1$ method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches $O(N_F^{-min\{r\alpha ,2-\alpha \}}+N_C^{-min\{2r\alpha ,4-2\alpha \}}+h_1^2+h_2^2)$, where $N_F$ and $N_C$ represent the number of the fine and coarse grids respectively, while $h_1$ and $h_2$ are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.