  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.
