Reduction in the Resonance Error in Numerical Homo

Authors: Antoine Gloria Zakaria Habibi

Publish Date: 2015/02/19

Volume: 16, Issue: 1, Pages: 217-296

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Abstract

This paper is the followup of Gloria Math Models Methods Appl Sci 2181601–1630 2011 One common drawback among numerical homogenization methods is the presence of the socalled resonance error which roughly speaking is a function of the ratio fracvarepsilon 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 nonnecessarily symmetric stationary ergodic coefficients We quantify this reduction for the class of periodic coefficients for the Kozlov subclass of almostperiodic coefficients and for the subclass of random coefficients that satisfy a spectral gap estimate eg 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

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