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We introduce a finite element method for numerical upscaling of second orderelliptic equations with highly heterogeneous coefficients. The method is basedon a mixed formulation of the problem and the concepts of the domaindecomposition and the hybrid discontinuous Galerkin methods. The methodutilizes three different scales: (1) the scale of the partition of the domainof the problem, (2) the sale of partition of the boundaries of the subdomains(related to the corresponding space of Lagrange multipliers), and (3) the finegrid scale that is assumed to resolve the scale of the heterogeneous variationof 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 coarsespaces. In this paper, we develop and study a multiscale HDG method that usespolynomial and homogenization-based multiscale spaces. These coarse spaces aredesigned for problems with scale separation. In our consequent paper, we planto extend our flexible HDG framework to more challenging multiscale problemswith non-separable scales and high contrast and consider enriched coarse spacesthat use appropriate local spectral problems.

A multiscale HDG method for second order elliptic equations. Part I. Polynomial and homogenization-based multiscale spaces