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

TL;DR

This paper introduces regularization and extrapolation techniques to significantly reduce resonance error in numerical homogenization, improving accuracy for various classes of coefficients and validating the approach through numerical experiments.

Contribution

It systematically applies regularization and extrapolation to minimize resonance error in homogenization, extending the analysis to non-symmetric and ergodic coefficients.

Findings

Resonance error reduction quantified for periodic, almost periodic, and random coefficients.

Numerical experiments in 2D demonstrate the method's efficiency.

Proves asymptotic consistency and provides quantitative estimates for periodic coefficients.

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.