A Fast Spatial Two-Grid Raviart-Thomas MFE Algorithm for the Nonlinear Time Fractional Diffusion Equations on Temporal Graded Mesh

Abstract

In this paper, a fast two-grid algorithm is constructed for solving the nonlinear time fractional diffusion equations by using the lowest order Raviart-Thomas mixed finite element (RTMFE) space and the temporal graded mesh. In the algorithm, the Caputo time fractional derivative is discretized by the well-known $L1$ formula on the graded mesh, the spatial domain is divided into coarse and fine grids, then a fast two steps algorithm is proposed by using two-grid computing method. The existence, uniqueness and unconditional stability for the proposed algorithm on the temporal graded mesh are derived in detail. In addition, when the analytical solution satisfies different regularity assumptions, the asymptotically optimal a priori error estimates in spatial direction are obtained on both the temporal uniform and graded meshes, which show that when spatial coarse and fine grid parameters satisfy $H=O(h^{1/2})$, the fast algorithm can obtain the same accuracy as the RTMFE algorithm. Finally, two numerical examples with different regularity conditions are provided to demonstrate the theoretical results.