  We propose some multigrid methods for solving the algebraic systems resulting
from finite element approximations of space fractional partial differential
equations (SFPDEs). It is shown that our multigrid methods are optimal, which
means the convergence rates of the methods are independent of the mesh size and
mesh level. Moreover, our theoretical analysis and convergence results do not
require regularity assumptions of the model problems. Numerical results are
given to support our theoretical findings.
