×

Reduction in the resonance error in numerical homogenization. II: Correctors and extrapolation. (English) Zbl 1335.65086

This paper provides the reader with a systematic study to regularization and extrapolation for reducing the resonance error in approximation of homogenized coefficients and correctors in the frame of non-necessarily symmetric stationary ergodic coefficients. The reduction approach is addressed for the class of periodic coefficients, for the Kozlov subclass of almost periodic coefficients and for random coefficients satisfying a special gap estimate such as Poisson random inclusions. Along with the theoretical results, several numerical experiments are also presented.
For Part I see [A. Gloria, Math. Models Methods Appl. Sci. 21, No. 8, 1601–1630 (2011; Zbl 1233.35016)].

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure

Citations:

Zbl 1233.35016
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] A. Abdulle. On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul., 4:447-459, 2005. · Zbl 1092.65093
[2] G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23:1482-1518, 1992. · Zbl 0770.35005
[3] G. Allaire and R. Brizzi. A multiscale finite element method for numerical homogenization. Multiscale Model. Simul., 4:790-812, 2005. · Zbl 1093.35007
[4] 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. · Zbl 1072.76560
[5] I. Babuska and R. Lipton. \[L^2\] L2-global to local projection: an approach to multiscale analysis. Math. Models Methods Appl. Sci., 21(11):2211-2226, 2011. · Zbl 1242.65234
[6] 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. · Zbl 0404.35001
[7] 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. · Zbl 1262.65115
[8] A. Bourgeat, A. Mikelić, and S. Wright. Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math., 456:19-51, 1994. · Zbl 0808.60056
[9] W. E, B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden. Heterogeneous multiscale methods: A review. Commun. Comput. Phys., 2:367-450, 2007. · Zbl 1164.65496
[10] 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. · Zbl 1060.65118
[11] Weinan E. Principles of multiscale modeling. Cambridge University Press, Cambridge, 2011. · Zbl 1238.00010
[12] 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. · Zbl 1156.74362
[13] 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. · Zbl 0951.65105
[14] 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. · Zbl 1315.60116
[15] FreeFEM. http://www.freefem.org/. · Zbl 1245.74049
[16] A. Gloria. An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. Multiscale Model. Simul., 5(3):996-1043, 2006. · Zbl 1119.74038
[17] A. Gloria. An analytical framework for numerical homogenization - Part II: windowing and oversampling. Multiscale Model. Simul., 7(1):275-293, 2008. · Zbl 1156.74362
[18] A. Gloria. Reduction of the resonance error - Part 1: Approximation of homogenized coefficients. Math. Models Methods Appl. Sci., 21(8):1601-1630, 2011. · Zbl 1233.35016
[19] A. Gloria. Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations. M2AN Math. Model. Numer. Anal., 46(1):1-38, 2012. · Zbl 1282.35038
[20] A. Gloria. Numerical homogenization: survey, new results, and perspectives. Esaim. Proc., 37, 2012. Mathematical and numerical approaches for multiscale problem. · Zbl 1329.65300
[21] A. Gloria and J.-C. Mourrat. Spectral measure and approximation of homogenized coefficients. Probab. Theory. Relat. Fields, 154(1), 2012. · Zbl 1264.35021
[22] 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. · Zbl 1307.35029
[23] 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. · Zbl 1314.39020
[24] A. Gloria and F. Otto. Quantitative results on the corrector equation in stochastic homogenization. arXiv:1409.0801. · Zbl 1387.35032
[25] P. Henning and D. Peterseim. Oversampling for the Multiscale Finite Element Method. Multiscale Model. Simul., 11(4):1149-1175, 2013. · Zbl 1297.65155
[26] 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. · Zbl 0880.73065
[27] 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. · Zbl 0922.65071
[28] 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. · Zbl 1085.65109
[29] V.V. Jikov, S.M. Kozlov, and O.A. Oleinik. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994. · Zbl 0801.35001
[30] S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients. Mat. Sb. (N.S.), 107(149)(2):199-217, 317, 1978. · Zbl 1085.65109
[31] H. Owhadi and L. Zhang. Metric-based upscaling. Comm. Pure Appl. Math., 60(5):675-723, 2007. · Zbl 1190.35070
[32] 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. · Zbl 1244.65140
[33] H. Owhadi, L. Zhang, and L. Berlyand. Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. ESAIM: Mathematical Modelling and Numerical Analysis, 2014. · Zbl 1296.41007
[34] 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.
[35] M. Vogelius. A homogenization result for planar, polygonal networks. RAIRO Modél. Math. Anal. Numér., 25(4):483-514, 1991. · Zbl 0737.35126
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.