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Corrector Analysis of a Heterogeneous Multi-scale Scheme forElliptic Equations with Random Potential

Published online by Cambridge University Press:  20 February 2014

Guillaume Bal  and
Wenjia Jing
Guillaume Bal
Affiliation:
Department of Applied Physics and Applied Mathematics, Columbia University, 10027 New York, USA.. gb2030@columbia.edu
Wenjia Jing
Affiliation:
Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75230 Paris Cedex 05, France.; wjing@dma.ens.fr
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Abstract

This paper analyzes the random fluctuations obtained by a heterogeneous multi-scalefirst-order finite element method applied to solve elliptic equations with a randompotential. Several multi-scale numerical algorithms have been shown to correctly capturethe homogenized limit of solutions of elliptic equations with coefficients modeled asstationary and ergodic random fields. Because theoretical results are available in thecontinuum setting for such equations, we consider here the case of a second-order ellipticequations with random potential in two dimensions of space. We show that the randomfluctuations of such solutions are correctly estimated by the heterogeneous multi-scalealgorithm when appropriate fine-scale problems are solved on subsets that cover the wholecomputational domain. However, when the fine-scale problems are solved over patches thatdo not cover the entire domain, the random fluctuations may or may not be estimatedaccurately. In the case of random potentials with short-range interactions, the varianceof the random fluctuations is amplified as the inverse of the fraction of the mediumcovered by the patches. In the case of random potentials with long-range interactions,however, such an amplification does not occur and random fluctuations are correctlycaptured independent of the (macroscopic) size of the patches. These results areconsistent with those obtained in [9] for moregeneral equations in the one-dimensional setting and provide indications on the loss inaccuracy that results from using coarser, and hence computationally less intensive,algorithms.

Keywords

Equations with random coefficientsmulti-scale finite element methodheterogeneous multi-scale methodcorrector testlong-range correlations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 387 - 409
Copyright
© EDP Sciences, SMAI, 2014

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References

Allaire, G. and Brizzi, R., A multiscale finite element method for numerical homogenization. Multiscale Model. Simul. 4 (2005) 790–812. (electronic). Google Scholar
Arbogast, T., Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal. 42 (2004) 576–598. Google Scholar
I. Babuska, Homogenization and its applications, mathematical and computational problems, Numerical Solutions of Partial Differential Equations-III, edited by B. Hubbard (SYNSPADE 1975, College Park, MD, May 1975). Academic Press, New York (1976) 89–116.
Babuska, I., Solution of interface by homogenization. I, II, III. SIAM J. Math. Anal. 7 (1976) 603–634, 635–645. Google Scholar
Babuska, I., Solution of interface by homogenization. I, II, III. SIAM J. Math. Anal. 8 (1977) 923–937. Google Scholar
Bal, G., Central limits and homogenization in random media. Multiscale Model. Simul. 7 (2008) 677–702. Google Scholar
Bal, G., Garnier, J., Gu, Y. and Jing, W., Corrector theory for elliptic equations with oscillatory and random potentials with long range correlations. Asymptot. Anal. 77 (2012) 123–145. Google Scholar
Bal, G., Garnier, J., Motsch, S. and Perrier, V., Random integrals and correctors in homogenization. Asymptot. Anal. 59 (2008) 1–26. Google Scholar
Bal, G. and Jing, W., Corrector theory for MsFEM and HMM in random media. Multiscale Model. Simul. 9 (2011) 1549–1587. Google Scholar
Bal, G. and Ren, K., Physics-based models for measurement correlations: application to an inverse Sturm-Liouville problem. Inverse Problems 25 (2009) 055006, 13. Google Scholar
Berlyand, L. and Owhadi, H., Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast. Arch. Ration. Mech. Anal. 198 (2010) 677–721. Google Scholar
Bourgeat, A. and Piatnitski, A., Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptot. Anal. 21 (1999) 303–315. Google Scholar
P.G. Ciarlet, The finite element method for elliptic problems, in vol. 4 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam (1978).
P. Doukhan, Mixing, Properties and examples, in vol. 85 of Lect. Notes Stat. Springer-Verlag, New York (1994).
Engquist, W.E.B., Li, X., Ren, W. and Vanden-Eijnden, E., Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2 (2007) 367–450. Google Scholar
Ming, W.E.P. and Zhang, P., Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc. 18 (2005) 121–156 (electronic). Google Scholar
Figari, R., Orlandi, E. and Papanicolaou, G., Mean field and Gaussian approximation for partial differential equations with random coefficients. SIAM J. Appl. Math. 42 (1982) 1069–1077. Google Scholar
Goldstein, C.I., Variational crimes and L ∞ error estimates in the finite element method. Math. Comp. 35 (1980) 1131–1157. Google Scholar
Hou, T.Y., Wu, X.-H. and Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913–943. Google Scholar
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag, New York (1994)
D. Khoshnevisan, Multiparameter processes. Springer Monographs in Mathematics. Springer-Verlag, New York (2002).An introduction to random fields.
Kozlov, S.M., The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188–202, 327. Google Scholar
E.H. Lieb and M. Loss, Analysis, in vol. 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2nd edn. (2001).
Nolen, J. and Papanicolaou, G., Fine scale uncertainty in parameter estimation for elliptic equations. Inverse Problems 25 (2009) 115021–115022. Google Scholar
Owhadi, H. and Zhang, L., Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast. Multiscale Model. Simul. 9 (2011) 1373–1398. Google Scholar
G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Random fields, vol. I, II (Esztergom, 1979). In vol. 27 of Colloq. Math. Soc. János Bolyai. North-Holland, Amsterdam (1981) 835–873.
Rannacher, R. and Scott, R., Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38 (1982) 437–445. Google Scholar
M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, Harcourt Brace Jovanovich Publishers, New York (1975).
Scott, R., Optimal L ∞ estimates for the finite element method on irregular meshes. Math. Comput. 30 (1976) 681–697. Google Scholar
Taqqu, M.S., Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 (1979) 53–83. Google Scholar
Wahlbin, L.B., Maximum norm error estimates in the finite element method with isoparametric quadratic elements and numerical integration. RAIRO Anal. Numér. 12 (1978) 173–202. Google Scholar