  In this article, we discuss the efficient implementation of powerful domain
decomposition smoothers for multigrid methods for high order discontinuous
Galerkin (DG) finite element methods. In particular, we study the inversion of
matrices associated to mesh cells and to the patches around a vertex,
respectively, in order to obtain fast local solvers for additive and
multiplicative subspace correction methods. The effort of inverting local
matrices for tensor product polynomials of degree $k$ is reduced from $\mathcal
O(k^{3d})$ to $\mathcal O(dk^{d+1})$ by exploiting the separability of the
differential operator and resulting low rank representation of its inverse as a
prototype for more general low rank representations.
