  Fractional diffusion equations (FDEs) are a mathematical tool used for
describing some special diffusion phenomena arising in many different
applications like porous media and computational finance. In this paper, we
focus on a two-dimensional space-FDE problem discretized by means of a second
order finite difference scheme obtained as combination of the Crank-Nicolson
scheme and the so-called weighted and shifted Gr\"unwald formula.
  By fully exploiting the Toeplitz-like structure of the resulting linear
system, we provide a detailed spectral analysis of the coefficient matrix at
each time step, both in the case of constant and variable diffusion
coefficients. Such a spectral analysis has a very crucial role, since it can be
used for designing fast and robust iterative solvers. In particular, we employ
the obtained spectral information to define a Galerkin multigrid method based
on the classical linear interpolation as grid transfer operator and
damped-Jacobi as smoother, and to prove the linear convergence rate of the
corresponding two-grid method. The theoretical analysis suggests that the
proposed grid transfer operator is strong enough for working also with the
V-cycle method and the geometric multigrid. On this basis, we introduce two
computationally favourable variants of the proposed multigrid method and we use
them as preconditioners for Krylov methods. Several numerical results confirm
that the resulting preconditioning strategies still keep a linear convergence
rate.
