  This paper considers the iterative solution of linear systems arising from
discretization of the anisotropic radiative transfer equation with
discontinuous elements on the sphere. In order to achieve robust convergence
behavior in the discretization parameters and the physical parameters we
develop preconditioned Richardson iterations in Hilbert spaces. We prove
convergence of the resulting scheme. The preconditioner is constructed in two
steps. The first step borrows ideas from matrix splittings and ensures mesh
independence. The second step uses a subspace correction technique to reduce
the influence of the optical parameters. The correction spaces are build from
low-order spherical harmonics approximations generalizing well-known diffusion
approximations. We discuss in detail the efficient implementation and
application of the discrete operators. In particular, for the considered
discontinuous spherical elements, the scattering operator becomes dense and we
show that $\mathcal{H}$- or $\mathcal{H}^2$-matrix compression can be applied
in a black-box fashion to obtain almost linear or linear complexity when
applying the corresponding approximations. The effectiveness of the proposed
method is shown in numerical examples.
