  It is known that the solution of a conservative steady-state two-sided
fractional diffusion problem can exhibit singularities near the boundaries. As
consequence of this, and due to the conservative nature of the problem, we
adopt a finite volume elements discretization approach over a generic
non-uniform mesh. We focus on grids mapped by a smooth function which consist
in a combination of a graded mesh near the singularity and a uniform mesh where
the solution is smooth. Such a choice gives rise to Toeplitz-like
discretization matrices and thus allows a low computational cost of the
matrix-vector product and a detailed spectral analysis. The obtained spectral
information is used to develop an ad-hoc parameter free multigrid
preconditioner for GMRES, which is numerically shown to yield good convergence
results in presence of graded meshes mapped by power functions that accumulate
points near the singularity. The approximation order of the considered graded
meshes is numerically compared with the one of a certain composite mesh given
in literature that still leads to Toeplitz-like linear systems and is then
still well-suited for our multigrid method. Several numerical tests confirm
that power graded meshes result in lower approximation errors than composite
ones and that our solver has a wide range of applicability.
