On the heterogeneous multiscale method with various macroscopic solvers

The heterogeneous multiscale method (HMM) is a general method for efficient numerical solution of problems with multiscales. It consists of two components: an overall macroscopic solver for macrovariables on a macrogrid and an estimation of the missing macroscopic data from the microscopic model. In this paper we present a state-of-the-art review of the HMM with various macroscopic solvers, including finite differences, finite elements, discontinuous Galerkin, mixed finite elements, control volume finite elements, nonconforming finite elements, and mixed covolumes. The first four solvers have been studied in the HMM setting; the others are not. As example, the HMM with the nonconforming finite element macroscopic solver for nonlinear and random homogenization problems is also studied here.