Comparison of Four Multiscale Methods for Elliptic Problems

Abstract

Four representative multiscale methods, namely asymptotic homogenization method (AHM), heterogeneous multiscale method (HMM), variational multiscale (VMS) method and multiscale finite element method (MsFEM), for elliptic problems with multiscale coefficients are surveyed. According to the features they possess, these methods are divided into two categories. AHM and HMM belong to the up–down framework. The feature of the framework is that the macroscopic solution is solved first with the help of effective information computed in local domains, and then the multiscale solution is resolved in local domains using the macroscopic solution when necessary. VMS method andMsFEM fall in the uncoupling framework. The feature of the framework is that the multiscale solution in the global domain is resolved directly making use of special functions defined in the macroscopic mesh. The cost of these multiscale methods is compared and their application to a functionally graded material is illustrated. Moreover, the manner to resolve multiscale solutions in AHM and HMM is compared, and it is found that AHM and HMM provide similar multiscale solutions.