Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation

We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann--Liouville type and order α∈ (1,2). We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank--Nicolson method. Error estimates in the L²(D)- and Hα/2(D)-norm are derived for the semidiscrete scheme and in the L²(D)-norm for the fully discrete schemes. These estimates cover 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.