A heterogeneous multiscale modeling framework for hierarchical systems of partial differential equations

This paper presents a heterogeneous multiscale method with efficient interscale coupling for scale-dependent phenomena modeled via a hierarchy of partial differential equations. Physics at the global level is governed by one set of partial differential equations, whereas features in the solution that are beyond the resolution capability of the coarser models are accounted for by the next refined set of differential equations. The proposed method seamlessly integrates different sets of equations governing physics at various levels, and represents a consistent top-down and bottom-up approach to multi-model modeling problems. For the top-down coupling of equations, this method provides a variational residual-based embedding of the response from the coarser or global system equations, into the corresponding local or refined system equations. To account for the effects of local phenomena on the global response of the system, the method also accommodates bottom-up embedding of the response from the local or refined mathematical models into the global or coarser model equations. The resulting framework thus provides a consistent way of coupling physics between disparate partial differential equations by means of up-scaling and down-scaling of the mathematical models. An integral aspect of the proposed framework is an uncertainty quantification and error estimation module. The structure of this error estimator is investisated and its mathematical implications are delineated.