  The Riesz maps of the $L^2$ de Rham complex frequently arise as subproblems
in the construction of fast preconditioners for more complicated problems. In
this work we present multigrid solvers for high-order finite element
discretizations of these Riesz maps with the same time and space complexity as
sum-factorized operator application, i.e.~with optimal complexity in polynomial
degree in the context of Krylov methods. The key idea of our approach is to
build new finite elements for each space in the de Rham complex with
orthogonality properties in both the $L^2$- and $H(\mathrm{d})$-inner products
($\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\})$ on the
reference hexahedron. The resulting sparsity enables the fast solution of the
patch problems arising in the Pavarino, Arnold--Falk--Winther and Hiptmair
space decompositions, in the separable case. In the non-separable case, the
method can be applied to an auxiliary operator that is sparse by construction.
With exact Cholesky factorizations of the sparse patch problems, the
application complexity is optimal but the setup costs and storage are not. We
overcome this with the finer Hiptmair space decomposition and the use of
incomplete Cholesky factorizations imposing the sparsity pattern arising from
static condensation, which applies whether static condensation is used for the
solver or not. This yields multigrid relaxations with time and space complexity
that are both optimal in the polynomial degree.

Numerical Analysis;Numerical Analysis