Communications in Mathematical Sciences
Volume 1 (2003)
The Heterognous Multiscale Methods
Pages: 87 – 132
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 , 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.