  We develop and analyze multilevel methods for nonuniformly elliptic operators
whose ellipticity holds in a weighted Sobolev space with an $A_2$--Muckenhoupt
weight. Using the so-called Xu-Zikatanov (XZ) identity, we derive a nearly
uniform convergence result, under the assumption that the underlying mesh is
quasi-uniform. We also consider the so-called $\alpha$-harmonic extension to
localize fractional powers of elliptic operators. Motivated by the scheme
proposed in [R.H. Nochetto, E. Otarola, and A.J. Salgado. A PDE approach to
fractional diffusion in general domains: a priori error analysis.
arXiv:1302.0698, 2013] we present a multilevel method with line smoothers and
obtain a nearly uniform convergence result on anisotropic meshes. Numerical
experiments reveal a competitive performance of our method.
