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...zes the microscopic and the macroscopic models in a domain-decomposition framework, becomes necessary, see for examples [15, 29, 31, 48, 57, 76, 80, 115, 90, 116] for multiscale kinetic problems, and =-=[38, 39, 77]-=- for broader areas of multiscale modeling and simulation. Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA. e-mail: jin@math.wisc.edu. Research was partially supporte...

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...dic media with scale separation or in the case of partial differential equations with sufficiently smooth coefficients 22• Multi-scale finite volume methods [56]. • Heterogeneous Multi-scale Methods =-=[30]-=-. • Wavelet based homogenization [43], [28], [23], [18], [7], [19]. • Residual free bubbles methods [20]. • Discontinuous enrichment methods [36], [35]. • Partition of Unity Methods [39]. • Energy Min...

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...fferential equations whose coefficients depend on a small parameter denoted by ε, without resolving all the details at the scale ε (see for instance [15, 16] for the multiscale finite element method, =-=[6, 5]-=- for the heterogeneous multiscale method, and [8] for a general analysis). In the case of periodic coefficients, it amounts to approximating the solution to the homogenized problem. To be more precise...

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...uch as the multi-grid method have given impetus to finding general framework for multiscale methods. Most notable among these are the systematic up-scaling approach of Brandt [2], the heterogeneous multiscale method [3–5] and the “equation-free” approach [6]. The hope is that a general framework might lead to general designing principles for multiscale methods and guidelines for carrying out error analysis, as was the case of finite difference and finite element methods for solving differential equations. To a large extent, 2 the heterogeneous multiscale method has indeed achieved these goals [4]. In Ref. [2], Achi Brandt reviewed a general strategy for extending the multigrid method and renormalization group analysis to multi-physics problems. The new strategy, which is now called “systematic up-scaling”, allows the use of discrete models such as Monte Carlo methods or molecular dynamics (besides traditional partial differential equation models) at the finest level of the calculation, and it does not require explicit macroscale models to begin with. In fact, Brandt remarked that one might be able to construct an effective macroscale model from the data accumulated during the computat...

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...ty analysis, the proof is concluded in Section 5. 1.4. Related works. Recently there are a lot of papers discussing various atomistic/continuum coupling strategies as summarized in the recent reviews =-=[10,15,29, 33]-=-, we will only mention some of the works that are closely related to ours and refer the readers to these reviews and the references therein. The hybrid method resembles several methods in the literatu...

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...temporary mathematically based approaches include the Multiscale FEM [13], [14], global changes of coordinates for upscaling porous media flows [10], [11] , the heterogeneous multiscale methods (HMM) =-=[8]-=-,[9], [12], an adaptive coarse scale - fine scale projection method [18], numerical homogenization methods for L∞ coefficients based on harmonic coordinates and elliptic inequalities [19], [20], [6], ...