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.
Similar content being viewed by others
Notes
The number of sub-/superscripts may not ease the reading, and we encourage the reader to simply consider \(H_1=H_2=\rho \) everywhere, although we shall need all the parameters in the proof.
References
A. Abdulle. On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul., 4:447–459, 2005.
G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23:1482–1518, 1992.
G. Allaire and R. Brizzi. A multiscale finite element method for numerical homogenization. Multiscale Model. Simul., 4:790–812, 2005.
T. Arbogast. Numerical subgrid upscaling of two-phase flow in porous media. In Numerical treatment of multiphase flows in porous media (Beijing, 1999), volume 552 of Lecture Notes in Phys., pages 35–49. Springer, Berlin, 2000.
I. Babuska and R. Lipton. \(L^2\)-global to local projection: an approach to multiscale analysis. Math. Models Methods Appl. Sci., 21(11):2211–2226, 2011.
A. Bensoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic analysis for periodic structures, volume 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1978.
X. Blanc and C. Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Netw. Heterog. Media, 5(1):1–29, 2010.
A. Bourgeat, A. Mikelić, and S. Wright. Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math., 456:19–51, 1994.
W. E, B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden. Heterogeneous multiscale methods: A review. Commun. Comput. Phys., 2:367–450, 2007.
W. E, P.B. Ming, and P.W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., 18:121–156, 2005.
Weinan E. Principles of multiscale modeling. Cambridge University Press, Cambridge, 2011.
Y. Efendiev and T. Y. Hou. Multiscale finite element methods, volume 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York, 2009. Theory and applications.
Y.R. Efendiev, T.Y. Hou, and X.H. Wu. Convergence of a nonconforming multiscale finite element method. SIAM J. Num. Anal., 37:888–910, 2000.
A.-C. Egloffe, A. Gloria, J.-C. Mourrat, and T. N. Nguyen. Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth. IMA J. Num. Anal., 2014. doi:10.1093/imanum/dru010.
FreeFEM. http://www.freefem.org/.
A. Gloria. An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. Multiscale Model. Simul., 5(3):996–1043, 2006.
A. Gloria. An analytical framework for numerical homogenization - Part II: windowing and oversampling. Multiscale Model. Simul., 7(1):275–293, 2008.
A. Gloria. Reduction of the resonance error - Part 1: Approximation of homogenized coefficients. Math. Models Methods Appl. Sci., 21(8):1601–1630, 2011.
A. Gloria. Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations. M2AN Math. Model. Numer. Anal., 46(1):1–38, 2012.
A. Gloria. Numerical homogenization: survey, new results, and perspectives. Esaim. Proc., 37, 2012. Mathematical and numerical approaches for multiscale problem.
A. Gloria and J.-C. Mourrat. Spectral measure and approximation of homogenized coefficients. Probab. Theory. Relat. Fields, 154(1), 2012.
A. Gloria, S. Neukamm, and F. Otto. An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. M2AN Math. Model. Numer. Anal., 2014. Special issue 2014: Multiscale problems and techniques.
A. Gloria, S. Neukamm, and F. Otto. Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math., 2014. DOI 10.1007/s00222-014-0518-z.
A. Gloria and F. Otto. Quantitative results on the corrector equation in stochastic homogenization. arXiv:1409.0801.
P. Henning and D. Peterseim. Oversampling for the Multiscale Finite Element Method. Multiscale Model. Simul., 11(4):1149–1175, 2013.
T.Y. Hou and X.H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134:169–189, 1997.
T.Y. Hou, X.H. Wu, and Z.Q. Cai. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput., 68:913–943, 1999.
T.Y. Hou, X.H. Wu, and Y. Zhang. Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Comm. in Math. Sci., 2(2):185–205, 2004.
V.V. Jikov, S.M. Kozlov, and O.A. Oleinik. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994.
S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients. Mat. Sb. (N.S.), 107(149)(2):199–217, 317, 1978.
H. Owhadi and L. Zhang. Metric-based upscaling. Comm. Pure Appl. Math., 60(5):675–723, 2007.
H. Owhadi and L. Zhang. Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast. Multiscale Model. Simul., 9(4):1373–1398, 2011.
H. Owhadi, L. Zhang, and L. Berlyand. Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. ESAIM: Mathematical Modelling and Numerical Analysis, 2014.
G.C. Papanicolaou and S.R.S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. In Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pages 835–873. North-Holland, Amsterdam, 1981.
M. Vogelius. A homogenization result for planar, polygonal networks. RAIRO Modél. Math. Anal. Numér., 25(4):483–514, 1991.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Philippe Ciarlet.
Rights and permissions
About this article
Cite this article
Gloria, A., Habibi, Z. Reduction in the Resonance Error in Numerical Homogenization II: Correctors and Extrapolation. Found Comput Math 16, 217–296 (2016). https://doi.org/10.1007/s10208-015-9246-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10208-015-9246-z
Keywords
- Numerical homogenization
- Resonance error
- Effective coefficients
- Correctors
- Periodic
- Almost periodic
- Random