Convergence and superconvergence analysis of finite element methods for the time fractional diffusion equation

In this paper, two classes of finite element methods (FEMs) for the two-dimensional time fractional diffusion equation (TFDE) with non-smooth solution are proposed and analyzed. In the temporal direction, we adopt nonuniform L2-1(sigma) method, and in the spatial direction, the conforming bilinear element and the nonconforming modified quasi-Wilson element are utilized. For the proposed conforming and nonconforming FEMs, by using new fractional discrete Gronwall inequalities, the theoretical analysis including the L-2-norm error estimates and H-1-norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global H-1-norm superconvergence results are presented. Finally, some numerical results illustrate the correctness of theoretical analysis. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.