Finite element method for space-time fractional diffusion equation

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