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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:
35B27Homogenization; equations in media with periodic structure (PDE)
35J20Second order elliptic equations, variational methods
74Q05Homogenization in equilibrium problems (solid mechanics)
76N25Flow control and optimization (compressible fluids and gas dynamics)
35F20General theory of first order nonlinear PDE
65J05General theory of numerical methods in abstract spaces
76M25Other numerical methods (fluid mechanics)
76M50Homogenization (fluid mechanics)
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