$L2-1_\sigma$ Finite Element Method for Time-Fractional Diffusion Problems with Discontinuous Coefficients

Abstract

A time-fractional diffusion equation with an interface problem caused by discontinuous coefficients is considered. To solve it, in the temporal direction Alikhanov's L2-1 σ method with graded mesh is presented to deal with the weak singularity at t = 0, while in the spatial direction a finite element method with uniform mesh is employed to handle the discontinuous coefficients. Then, with the help of discrete fractional Grönwall inequality and the robustness theory of α → 1 − , we show that the method has stable error bounds at α → 1 − , the fully discrete schemes L 2 (Ω) norm and H 1 (Ω) semi-norm are unconditionally stable, and the optimal convergence order is (h 2 + N − min{rα,2} ) and (h + N − min{rα,2} ), respectively, where, h, N , α, r is the total number of spatial parameter, the time-fractional order coefficient, and the time grid constant. Finally, three numerical examples are provided to illustrate our theoretical results.