E, Weinan; Engquist, Bjorn The heterogeneous multiscale methods. (English) Zbl 1093.35012 Commun. Math. Sci. 1, No. 1, 87-132 (2003). 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. Reviewer: Marco Codegone (Torino) Cited in 10 ReviewsCited in 460 Documents 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 Keywords:heterogeneous media; homogenization; multiscale methods; molecular dynamics; kinetic models; interfacial dynamics × Cite Format Result Cite Review PDF Full Text: DOI Euclid