Two-grid Algorithm For A Class Of Time-fractional Nonlinear Parabolic Problems

In this paper,a two-grid finite element method for solving a class of time-fractional nonlinear parabolic equations is presented.Firstly,the Caputo time-fractional derivative is discretized by the L1 method,and then the standard fully discrete finite element problem for the time-fractional nonlinear parabolic equation is obtained,the stability and error estimate of the fully discrete problem under L2 and H1 norms are obtained,and the theoretical results are verified by a numerical experiment.Secondly,on the basis of the standard finite element method,we design the corresponding two-grid algorithm,that is,solving the nonlinear problem on the coarse finite element space SH(? Sh),and an approximate linear problem is solved in the fine finite element space Sh.Then,the stability and error estimate of the two-grid algorithm are analyzed.The theoretical results show that the algorithm is as stable as the standard fully discrete finite element method if the size H of the coarse grid and the size h of the fine grid satisfies H=O(hr-1/r)(r≥1+d/2,where d=1,2,3),finally,the theoretical results of the two-grid algorithm are verified by the numerical experiment.