  We present a polynomial multigrid method for nodal interior penalty and local
discontinuous Galerkin formulations of the Poisson equation on Cartesian grids.
For smoothing we propose two classes of overlapping Schwarz methods. The first
class comprises element-centered and the second face-centered methods. Within
both classes we identify methods that achieve superior convergence rates, prove
robust with respect to the mesh spacing and the polynomial order, at least up
to ${P=32}$. Consequent structure exploitation yields a computational
complexity of $O(PN)$, where $N$ is the number of unknowns. Further we
demonstrate the suitability of the face-centered method for element aspect
ratios up to 32.
