  The nonlocality of the fractional operator causes numerical difficulties for
long time computation of the time-fractional evolution equations. This paper
develops a high-order fast time-stepping discontinuous Galerkin finite element
method for the time-fractional diffusion equations, which saves storage and
computational time. The optimal error estimate $O(N^{-p-1} + h^{m+1} +
\varepsilon N^{r\alpha})$ of the current time-stepping discontinuous Galerkin
method is rigorous proved, where $N$ denotes the number of time intervals, $p$
is the degree of polynomial approximation on each time subinterval, $h$ is the
maximum space step, $r\ge1$, $m$ is the order of finite element space, and
$\varepsilon>0$ can be arbitrarily small. Numerical simulations verify the
theoretical analysis.
