Fast time-stepping discontinuous Galerkin method for the subdiffusion equation

The nonlocality of the fractional operator causes numerical difficulties for long time computation for time-fractional evolution equations. This paper develops a high-order fast time-stepping discontinuous Galerkin (DG) finite element method for a time-fractional diffusion equation, which saves storage and computational time. An optimal error estimate of the form $O(N^{-p-1} + h^{m+1} + \varepsilon N^{r\alpha })$ for the proposed time-stepping DG method is rigorously proved, where $N$ denotes the number of time intervals, $p$ is the degree of polynomial approximation on each time subinterval, $h$ is the maximum spatial mesh size, $r\ge 1$, $m$ is the order of finite element space and $\varepsilon>0$ can be arbitrarily small. Numerical simulations verify the theoretical analysis.