Numerical Algorithms

, Volume 72, Issue 3, pp 749–767

# Finite element method for space-time fractional diffusion equation

• L. B. Feng
• P. Zhuang
• F. Liu
• I. Turner
• Y. T. Gu
Original Paper

## Abstract

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ 2−γ + h 2) and O(τ 2 + h 2), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ 2 + h 2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.

### Keywords

Finite element method Space-time fractional diffusion equation Riesz derivative Caputo derivative Riemann-Liouville derivative

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## Authors and Affiliations

• L. B. Feng
• 1
• P. Zhuang
• 1
• 4
• F. Liu
• 2
• I. Turner
• 2
• Y. T. Gu
• 3
1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
2. 2.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
3. 3.School of Chemistry, Physics and Mechanical EngineeringQueensland University of TechnologyBrisbaneAustralia
4. 4.Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputationXiamen UniversityXiamenChina