A two-grid fully discrete Galerkin finite element approximation for fully nonlinear time-fractional wave equations

We construct a fully discrete implicit finite element scheme with L 1 method for solving fully nonlinear time-fractional wave equations, which is first considered in this work, and then we prove that this scheme is stable and derive the optimal error estimates. To improve the computational efficiency, a two-grid algorithm is developed based on the fully discrete finite element scheme. The stability is discussed and the optimal convergence rate O (h+H-2 +tau(3-beta)) in H1-norm is derived when the coarse-grid with mesh size H and the fine-grid with mesh size h satisfy H-2 = O(h), where tau is the time step size. A significant advantage is that this technique not only saves a lot of time but also maintains the numerical accuracy. Moreover, numerical examples are presented to demonstrate our theoretical results and to test the performance of the proposed algorithm.