  This paper is devoted to the multigrid convergence analysis for the linear
systems arising from the conforming linear finite element discretization of the
second order elliptic equations with anisotropic diffusion. The multigrid
convergence behavior is known to strongly depend on whether the discretization
grid is aligned or non-aligned with the anisotropic direction and analyses in
the paper will be mainly focused on two-level algorithms. For an aligned grid
case, a lower bound is given for point-wise smoother which shows deterioration
of convergence rate. In both aligned and non-aligned cases we show that for a
specially designed block smoother the convergence is uniform with respect to
both anisotropy ratio and mesh size in the energy norm. The analysis is
complemented with numerical experiments which confirm the theoretical results
