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The heterogeneous multiscale method, is presented as a general methodology for an efficient numerical computation of problems with multiple scales. The method relies on an efficient coupling between the macroscopic and microscopic models. In case the macroscopic model is not explicitly available or is invalid in part of the domain, the microscopic model is used to supply the necessary data for the macroscopic model. Scale separation is exploited so that coarse-grained variables can be evolved on macroscopic spatial/temporal scales using data that are predicted based on the simulation of the microscopic process on microscale spatial/temporal domains. Applications to homogenization, dislocation dynamics and crack propagation are discussed.
Weinan et al. (2003) studied this question.