Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh–Stokes equations

In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh–Stokes equations. The proposed two-grid FEM uses the L2-$1_{\sigma }$ scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The $L^2$-norm and $H^1$-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy $H=h^{\frac{1}{2}}$ and $H=h^{\frac{r}{2r+2}}$ respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-$1_{\sigma }$ scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.