Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes

Abstract

In this paper, an unconditionally stable fully discrete numerical scheme for the two-dimensional (2D) time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The L2-1σ scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization. The unconditional stability and convergence of the proposed scheme are proved rigorously. It is shown that the order O(h2+N−min{rα,2}) can be achieved, where h is the spatial step, N is the number of partition in temporal direction, r is the temporal meshes grading parameter and α is the order of fractional derivative. A numerical example is provided to verify the sharpness of our error analysis.