Fast time-stepping discontinuous Galerkin method for the subdiffusion equation

Abstract

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 $O(N^{-p-1}+h^{m+1}+\epsilon N^{r\alpha })$ 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, $r\ge 1$, $m$ is the order of finite element space, and $\epsilon >0$ can be arbitrarily small. Numerical simulations verify the theoretical analysis.

21 pages, 1 figure,4 tables