  In this paper, we design preconditioners for the matrix-free solution of
high-order continuous and discontinuous Galerkin discretizations of elliptic
problems based on FEM-SEM equivalence and additive Schwarz methods. The
high-order operators are applied without forming the system matrix, making use
of sum factorization for efficient evaluation. The system is preconditioned
using a spectrally equivalent low-order ($p=1$) finite element operator
discretization on a refined mesh. The low-order refined mesh is anisotropic and
not shape regular in the polynomial degree of the high-order operator,
requiring specialized solvers to treat the anisotropy. We make use of an
element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing.
The preconditioner is parallelized through an overlapping additive Schwarz
method that is robust in $h$ and $p$. The method is extended to interior
penalty and BR2 discontinuous Galerkin discretizations, for which it is also
robust in the size of the penalty parameter. Numerical results are presented on
a variety of examples, verifying the uniformity of the preconditioner.
