Two-grid finite element methods for nonlinear time-fractional parabolic equations

In this paper, an efficient two-grid finite element algorithm is proposed for solving time-fractional nonlinear parabolic equations. We first obtain the stability and error estimates of the standard fully discrete L1 finite element method by using the Gronwall inequalities in L2 and H1 norms. Based on the standard method, we design the corresponding two-grid algorithm and analyze its stability and error estimates. It is shown that this algorithm is as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy $H=O\left (h^{\frac {r-1}{r}}\right )(r\ge 1+\frac {d}{2},$ where d = 1,2,3). The theoretical results are illustrated by applying the proposed method to an numerical example.