The Heterognous Multiscale Methods

Abstract

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