Two-grid mixed finite element method combined with the BDF2-θ for a two-dimensional nonlinear fractional pseudo-hyperbolic wave equation

In this article, a fast two-grid mixed finite element (T-GMFE) algorithm based on a time second-order discrete scheme with parameter $\theta$ is considered to numerically solve a class of two-dimensional nonlinear fractional pseudo-hyperbolic wave models. The weighted and shifted Grünwald difference (WSGD) formula is used to approximate the fractional time derivative at time $t_{n-\theta }$, and the spatial direction is approximated by a two-grid $H^1$-Galerkin MFE method. The error estimates in both $L^2$ and $H^1$-norm for the fully discrete T-GMFE system are proved. Further, a modified T-GMFE scheme is proposed and the optimal error results are provided. Finally, computing results show the presented T-GMFE method can save computing time and improve the computational efficiency.