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The heterogeneous multiscale methods. (English) Zbl 1093.35012

A general methodology is presented to discuss and develop the heterogeneous multiscale method for the numerical computation of problems on heterogeneous media. The traditional approach for such problems is to obtain either analytically or empirically explicit equations for the scale of interest, eliminating other scales in the problem. In this paper, the authors present a general framework for designing and analysing numerical methods that deal with variational and dynamic problems. An efficient usage is made of the macroscopic and microscopic formulations to cover many of the existing methods but also to derive new technique from the general formulation. Application to problems such as homogenization, molecular dynamics, kinetic models and interfacial dynamics are discussed.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J20 Variational methods for second-order elliptic equations
74Q05 Homogenization in equilibrium problems of solid mechanics
76N25 Flow control and optimization for compressible fluids and gas dynamics
35F20 Nonlinear first-order PDEs
65J05 General theory of numerical analysis in abstract spaces
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M50 Homogenization applied to problems in fluid mechanics
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