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