  A new iterative method for non-LTE multilevel polarized radiative transfer in
hydrogen lines is presented. Iterative methods (such as the Jacobi method) tend
to damp out high-frequency components of the error fast, but converges poorly
due to slow reduction of low-frequency components. The idea is to use a set of
differently coarsed grids to reduce both the short- and long-period errors.
This leads to the so-called multigrid (MG) methods. For the grid of $N$ spatial
points, the number of iterations required to solve a non-LTE transfer problem
is of the order of O(N). This fact could be of great importance for problems
with fine structure and for multi-dimensional models. The efficiency of the
so-called standard MG iteration in comparison to Jacobi iteration is shown. The
formalism of density matrix is applied to the demonstrative example of 1D,
semi-infinite, non-magnetic, 3-principal level hydrogen atmospheric model. The
effect of depolarizing collisions with thermal electrons is taken into account
as well as general treatment of overlapping profiles.
