Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

Abstract

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0 < α < 1). The proposed method is based on high-order FEM for spacial discretization and finite difference method for time discretization. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h ^r+1 + τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.

Keywords:

• time-fractional partial differential equation
• finite element method
• Grwünwald–Letnikov derivative
• convergence rate
• error estimate

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestion and comments which have improved this article. This work is supported by the NSFXJ (No.2012211B21).