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The heterogenous multiscale method (HMM) is presented as a general for the efficient numerical computation of problems with and multiphysics on multigrids. Both variational and problems are considered. The method relies on an efficent between the macroscopic and microscopic models. In cases the macroscopic model is not explicity available or invalid, microscopic solver is used to supply the necessary data for the solver. Besides unifying several existing multiscale such as the ab initio molecular dynamics 13, methods 73, 69, 68 and projective methods for systems multiscales 34, 35, HMM also provides a methodology for new methods for a large variety of multiscale problems. A is presented for the analysis of the stability and accuracy HMM. Applications to problems such as homogenization, molecular, kinetic models and interfacial dynamics are discussed.
Weinan et al. (Wed,) studied this question.