Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

In this article, a second-order time discrete algorithm with a shifted parameter $\theta$ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $\theta$ (BDF2- $\theta$) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $t_{n-\theta }$, the $H^1$-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $L^2$-norm and $H^1$-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.