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

Fast time-stepping discontinuous Galerkin method for the subdiffusion equation