Heterogeneous multiscale methods: a review.

*(English)*Zbl 1164.65496Summary: 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.

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

74Q05 | Homogenization in equilibrium problems of solid mechanics |

74Q20 | Bounds on effective properties in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |