Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $\alpha \in (0,1)$ and $\beta \in (1,2)$. The global convergence order of the two-grid scheme reaches $O({\tau }^{3-\beta }+h^{r+1}+H^{2r+2})$, where $\tau$, $H$ and $h$ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.