  We introduce a finite element method for numerical upscaling of second order
elliptic equations with highly heterogeneous coefficients. The method is based
on a mixed formulation of the problem and the concepts of the domain
decomposition and the hybrid discontinuous Galerkin methods. The method
utilizes three different scales: (1) the scale of the partition of the domain
of the problem, (2) the sale of partition of the boundaries of the subdomains
(related to the corresponding space of Lagrange multipliers), and (3) the fine
grid scale that is assumed to resolve the scale of the heterogeneous variation
of the coefficients. Our proposed method gives a flexible framework that (1)
couples independently generated multiscale basis functions in each coarse patch
(2) provides a stable global coupling independent of local discretization,
physical scales and contrast (3) allows avoiding any constraints on coarse
spaces. In this paper, we develop and study a multiscale HDG method that uses
polynomial and homogenization-based multiscale spaces. These coarse spaces are
designed for problems with scale separation. In our consequent paper, we plan
to extend our flexible HDG framework to more challenging multiscale problems
with non-separable scales and high contrast and consider enriched coarse spaces
that use appropriate local spectral problems.
