Error Analysis of Finite Element Methods for Space-Fractional Parabolic Equations

We consider an initial/boundary value problem for one-dimensional fractional-order parabolic equations with a space fractional derivative of Riemann-Liouville type and order $\alpha\in (1,2)$. We study a spatial semidiscrete scheme with the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and Crank-Nicolson method. Error estimates in the $L^2\II$- and $H^{\alpha/2}\II$-norm are derived for the semidiscrete scheme, and in the $L^2\II$-norm for the fully discrete schemes. These estimates are for both smooth and nonsmooth initial data, and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.