  In this work, we propose an efficient finite element method for solving
fractional Sturm-Liouville problems involving either the Caputo or
Riemann-Liouville derivative of order $\alpha\in(1,2)$ on the unit interval
$(0,1)$. It is based on novel variational formulations of the eigenvalue
problem. Error estimates are provided for the finite element approximations of
the eigenvalues. Numerical results are presented to illustrate the efficiency
and accuracy of the method. The results indicate that the method can achieve a
second-order convergence for both fractional derivatives, and can provide
accurate approximations to multiple eigenvalues simultaneously.
