  We consider a standard elliptic partial differential equation and propose a
geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for
hybridized high-order finite element methods. The proposed unified approach is
applicable to any locally conservative hybridized finite element method
including multinumerics with different hybridized methods in different parts of
the domain. For these methods, the linear system involves only the unknowns
residing on the mesh skeleton, and constructing intergrid transfer operators is
therefore not trivial. The key to our geometric multigrid algorithm is the
physics-based energy-preserving intergrid transfer operators which depend only
on the fine scale DtN maps. Thanks to these operators, we completely avoid
upscaling of parameters and no information regarding subgrid physics is
explicitly required on coarse meshes. Moreover, our algorithm is
agglomeration-based and can straightforwardly handle unstructured meshes. We
perform extensive numerical studies with hybridized mixed methods, hybridized
discontinuous Galerkin method, weak Galerkin method, and a hybridized version
of interior penalty discontinuous Galerkin methods on a range of elliptic
problems including subsurface flow through highly heterogeneous porous media.
We compare the performance of different smoothers and analyze the effect of
stabilization parameters on the scalability of the multigrid algorithm.
