A hierarchical multiscale framework for problems with multiscale source terms

This paper presents a hierarchical multiscale framework for problems that involve multiscale source terms. An assumption on the additive decomposition of the source function results in consistent decoupling of the fully coupled system and constitutes the new method. The structure of this decomposition is investigated and its mathematical implications are delineated. This method results in variational embedding of fine-scale information that is derived from the fine-scale equations, in the corresponding coarse-scale equations. It therefore provides a mathematically consistent way of bridging information between disparate spatial scales in the response function that are induced by multiscale forcing functions.