Reduction of the resonance error in numerical homogenisation II: correctors and extrapolation

Antoine Gloria 1, 2 Zakaria Habibi 2
2 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : This paper is the companion article of [Gloria, M3AS, 21 (2011), No. 3, pp 1601-1630]. 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.
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Article dans une revue
Foundations of Computational Mathematics, Springer Verlag, 2016, 16 (1), pp.217--296. 〈10.1007/s10208-015-9246-z〉
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Antoine Gloria, Zakaria Habibi. Reduction of the resonance error in numerical homogenisation II: correctors and extrapolation. Foundations of Computational Mathematics, Springer Verlag, 2016, 16 (1), pp.217--296. 〈10.1007/s10208-015-9246-z〉. 〈hal-00933234v2〉

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