  In this paper we analyze the convergence properties of two-level and W-cycle
multigrid solvers for the numerical solution of the linear system of equations
arising from hp-version symmetric interior penalty discontinuous Galerkin
discretizations of second-order elliptic partial differential equations on
polygonal/polyhedral meshes. We prove that the two-level method converges
uniformly with respect to the granularity of the grid and the polynomial
approximation degree p, provided that the number of smoothing steps, which
depends on p, is chosen sufficiently large. An analogous result is obtained for
the W-cycle multigrid algorithm, which is proved to be uniformly convergent
with respect to the mesh size, the polynomial approximation degree, and the
number of levels, provided the number of smoothing steps is chosen sufficiently
large. Numerical experiments are presented which underpin the theoretical
predictions; moreover, the proposed theoretical assumptions are not fully
satisfied.
