  We present W-cycle multigrid algorithms for the solution of the linear system
of equations arising from a wide class of $hp$-version discontinuous Galerkin
discretizations of elliptic problems. Starting from a classical framework in
multigrid analysis, we define a smoothing and an approximation property, which
are used to prove the uniform convergence of the W-cycle scheme with respect to
the granularity of the grid and the number of levels. The dependence of the
convergence rate on the polynomial approximation degree $p$ is also tracked,
showing that the contraction factor of the scheme deteriorates with increasing
$p$. A discussion on the effects of employing inherited or non-inherited
sublevel solvers is also presented. Numerical experiments confirm the
theoretical results.
