Finite difference heterogeneous multi-scale method for homogenization problems

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The heterogeneous multiscale method* (2012, unknown): “Then D needs to be evaluated at the cell boundaries (Abdulle and E 2003).”
Analysis of the heterogeneous multiscale method for elliptic homogenization problems (2004, methodology): “For example, the finite difference method and the discontinuous Galerkin method have been employed as the macroscopic solver in [1] and [9], respectively.”
Heterogeneous Multiscale Methods: A Review (2007, methodology): “This is a simplified version of the methods presented in [3].”
Asymptotic and numerical homogenization (2008, methodology): “This is a simplified version of the methods presented in Abdulle and E (2003). Similar ideas can also be found in Durlofsky (1991). As the macro-scale solver, we choose a finite volume method on a macroscale grid, and we will let ∆x,∆y be the grid size.”