  We consider a boundary value problem involving a Riemann-Liouville fractional
derivative of order $\alpha\in (3/2,2)$ on the unit interval $(0,1)$. The
standard Galerkin finite element approximation converges slowly due to the
presence of singularity term $x^{\alpha-1}$ in the solution representation. In
this work, we develop a simple technique, by transforming it into a
second-order two-point boundary value problem with nonlocal low order terms,
whose solution can reconstruct directly the solution to the original problem.
The stability of the variational formulation, and the optimal regularity pickup
of the solution are analyzed. A novel Galerkin finite element method with
piecewise linear or quadratic finite elements is developed, and $L^2(D)$ error
estimates are provided. The approach is then applied to the corresponding
fractional Sturm-Liouville problem, and error estimates of the eigenvalue
approximations are given. Extensive numerical results fully confirm our
theoretical study.

Numerical Analysis