Finite element method for space-time fractional diffusion equation

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 s (1 <s ≤ 2), and the first order time derivative by a Caputo fractional derivative of order ? (0 <? ≤ 1). For the 0 <? < 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(?2?? + h2) and O(?2 + h2), respectively, in which ? is the time step size and h is the space step size. And for the case ? = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(?2 + h2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.