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Analysis of a onedimensional nonlocal quasicontinuum method
 SIMULATION 7(4), 1838–1875 (2009). DOI 10.1137/080725842. URL HTTP://LINK.AIP.ORG/LINK/?MMS/7/1838/1
"... The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the qu ..."
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Cited by 55 (4 self)
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The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the quasicontinuum method with uniform accuracy.
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review
, 2010
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Metric based upscaling
 Communications on Pure and Applied Mathematics
, 2007
"... We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the med ..."
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Cited by 23 (2 self)
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We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators. 1 Introduction and main results Let Ω be a bounded and convex domain of class C2. We consider the following benchmark PDE
REDUCTION OF THE RESONANCE ERROR PART 1: APPROXIMATION OF HOMOGENIZED COEFFICIENTS
, 2010
"... Abstract. This paper is concerned with the approximation of effective coefficients in homogenization of linear elliptic equations. One common drawback among numerical homogenization methods is the presence of the socalled resonance error, which roughly speaking is a function of the ratio ε/η, where ..."
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Cited by 21 (3 self)
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Abstract. This paper is concerned with the approximation of effective coefficients in homogenization of linear elliptic equations. One common drawback among numerical homogenization methods is the presence of the socalled resonance error, which roughly speaking is a function of the ratio ε/η, where η is a typical macroscopic lengthscale and ε is the typical size of the heterogeneities. In the present work, we propose an alternative for the computation of homogenized coefficients (or more generally a modified cellproblem), which is a first brick in the design of effective numerical homogenization methods. We show that this approach drastically reduces the resonance error in some standard cases.
Macroscopic fluid models with localized kinetic upscaling effects
 SIMUL
"... This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling ..."
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Cited by 20 (6 self)
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This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the nonequilibrium part of the distribution function. This equation is solved only locally and is related to the fluid equation through a downscaling effect. The method does not need to find an interface condition as do usual domain decomposition methods to match fluid and kinetic representations. We show our approach applies to problems that have a hydrodynamic time scale as well as to problems with diffusion time scale. Simple numerical schemes are proposed to discretized our models, and several numerical examples are used to validate the method.
A general strategy for designing seamless multiscale methods
 J. Comput. Phys
, 2009
"... Abstract We present a new general framework for designing multiscale methods. Compared with previous work such as Brandt's systematic upscaling, the heterogeneous multiscale method and the "equationfree" approach, this new framework has the distinct feature that it does not require ..."
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Cited by 16 (1 self)
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Abstract We present a new general framework for designing multiscale methods. Compared with previous work such as Brandt's systematic upscaling, the heterogeneous multiscale method and the "equationfree" approach, this new framework has the distinct feature that it does not require reinitializing the microscale model at each macro time step or each macro iteration step. In the new strategy, the macroand micromodels evolve simultaneously using different time steps (and therefore different clocks), and they exchange data at every step. The micromodel uses its own appropriate time step. The macromodel runs at a slower pace than required by accuracy and stability considerations for the macroscale dynamics, in order for the micromodel to relax. Examples are discussed and application to modeling complex fluids is presented.
Effectiveness of implicit methods for stiff stochastic differential equations
 Commun. Comput. Phys
, 2000
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Convergence of a forcebased hybrid method for atomistic and continuum models in three dimension
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Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems∗
"... The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in L∞(Ω). This class of coefficients includes as exam ..."
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Cited by 12 (1 self)
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The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in L∞(Ω). This class of coefficients includes as examples media with microstructure as well as media with multiple nonseparated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in [5], and elaborated in [3], [4] and [25]. The GFEM is constructed by partitioning the computational domain Ω into a collection of preselected subsets ωi, i = 1, 2,..m and constructing finite dimensional approximation spaces Ψi over each subset using local information. The notion of the Kolmogorov nwidth is used to identify the optimal local approximation spaces. These spaces deliver local approximations with errors that decay almost exponentially with the degrees of freedom Ni in the energy norm over ωi. The local spaces Ψi are used within the GFEM scheme to produce a finite dimensional subspace S N of H1(Ω) which is then employed in the Galerkin method. It is shown that the error in the Galerkin approximation decays in the energy norm almost exponentially ( i.e., superalgebraicly) with respect to the degrees of freedom N. When length scales “separate ” and the microstructure is sufficiently fine with respect to the length scale of the domain ωi it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the preasymtotic regime.