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

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 scale 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, and (3) allows avoiding any constraints [Arbogast et al., Multiscale Model. Simul., 6 (2007), pp. 319-346] 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 subsequent paper, we plan to extend our flexible HDG framework to more challenging multiscale problems with nonseparable scales and high contrast and consider enriched coarse spaces that use appropriate local spectral problems.