Fast time-stepping discontinuous Galerkin method for the subdiffusion equation

TL;DR

This paper introduces a high-order, fast time-stepping discontinuous Galerkin method for solving time-fractional diffusion equations, effectively reducing computational cost while maintaining accuracy.

Contribution

It develops a novel high-order fast time-stepping discontinuous Galerkin method with rigorous error estimates for time-fractional diffusion equations.

Findings

Optimal error estimate proved for the method

Numerical simulations confirm theoretical accuracy

Method reduces storage and computational time

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 $O(N^{-p-1} + h^{m+1} + \varepsilon N^{r\alpha})$ 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.