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Foundations of Computational Mathematics

, Volume 16, Issue 1, pp 217–296 | Cite as

Reduction in the Resonance Error in Numerical Homogenization II: Correctors and Extrapolation

  • Antoine GloriaEmail author
  • Zakaria Habibi
Article

Abstract

This paper is the follow-up of Gloria (Math Models Methods Appl Sci 21(8):1601–1630, 2011). One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the ratio \(\frac{\varepsilon }{\rho }\), where \(\rho \) is a typical macroscopic lengthscale and \(\varepsilon \) is the typical size of the heterogeneities. In the present work, we make a systematic use of regularization and extrapolation to reduce this resonance error at the level of the approximation of homogenized coefficients and correctors for general non-necessarily symmetric stationary ergodic coefficients. We quantify this reduction for the class of periodic coefficients, for the Kozlov subclass of almost-periodic coefficients, and for the subclass of random coefficients that satisfy a spectral gap estimate (e.g., Poisson random inclusions). We also report on a systematic numerical study in dimension 2, which demonstrates the efficiency of the method and the sharpness of the analysis. Last, we combine this approach to numerical homogenization methods, prove the asymptotic consistency in the case of locally stationary ergodic coefficients, and give quantitative estimates in the case of periodic coefficients.

Keywords

Numerical homogenization Resonance error Effective coefficients Correctors Periodic Almost periodic Random 

Mathematics Subject Classification

35J15 35B27 65N12 65N15 65B05 

Notes

Acknowledgments

The first author acknowledges financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).

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Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Université Libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.Project-Team MEPHYSTOInria Lille - Nord EuropeVilleneuve d’AscqFrance

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