  We present a polynomial multigrid method for the nodal interior penalty
formulation of the Poisson equation on three-dimensional Cartesian grids. Its
key ingredient is a weighted overlapping Schwarz smoother operating on
element-centered subdomains. The MG method reaches superior convergence rates
corresponding to residual reductions of about two orders of magnitude within a
single V(1,1) cycle. It is robust with respect to the mesh size and the ansatz
order, at least up to ${P=32}$. Rigorous exploitation of tensor-product
factorization yields a computational complexity of $O(PN)$ for $N$ unknowns,
whereas numerical experiments indicate even linear runtime scaling. Moreover,
by allowing adjustable subdomain overlaps and adding Krylov acceleration, the
method proved feasible for anisotropic grids with element aspect ratios up to
48.
