Finite element algorithm based on high-order time approximation for time fractional convection-diffusion equation

In this paper, finite element method with high-order approximation for time fractional derivative is considered and discussed to find the numerical solution of time fractional convection-diffusion equation. Some lemmas are introduced and proved, further the stability and error estimates are discussed and analyzed, respectively. The convergence result $O(h^{r+1}+tau^{3-alpha})$ can be derived, which illustrates that time convergence rate is higher than the order $(2-alpha)$ derived by $L1$-approximation. Finally, to validate our theoretical results, some computing data are provided.