Commun. Comput. Phys., 2 (2007), pp. 367-450.

Heterogeneous Multiscale Methods: A Review

Weinan E 1*, Bjorn Engquist 2, Xiantao Li 3, Weiqing Ren 4, Eric Vanden-Eijnden 4

1 Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA.
2 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA.
3 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.
4 Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.

Received 20 August 2006; Accepted 31 August 2006
Available online 5 October 2006


This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, micro-fluidics, solids, interface problems, stochastic problems, and statistically self-similar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

AMS subject classifications: 65N30, 74Q05, 74Q20, 39A12

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Key words: Multi-scale modeling, heterogeneous multi-scale method, multi-physics models, constrained micro-scale solver, data estimation.

*Corresponding author.
Email: (W. E), (B. Engquist), (X. Li), (W. Ren), (E. Vanden-Eijnden)

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