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

Arif Masud, Guglielmo Scovazzi

Research output: Contribution to journalArticlepeer-review


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 investigated and its mathematical implications are delineated.

Original languageEnglish (US)
Pages (from-to)28-42
Number of pages15
JournalInternational Journal for Numerical Methods in Fluids
Issue number1-3
StatePublished - Jan 2011


  • Heterogeneous methods
  • Hierarchical models
  • Interscale coupling
  • Multiscale methods
  • Partial differential equations
  • Variational methods

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Computer Science Applications
  • Applied Mathematics


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