Free Access
Issue
ESAIM: M2AN
Volume 48, Number 3, May-June 2014
Page(s) 815 - 858
DOI https://doi.org/10.1051/m2an/2013122
Published online 08 April 2014
  1. J. Aarnes and Y.R. Efendiev, Mixed multiscale finite element methods for stochastic porous media flows. SIAM J. Sci. Comput. 30 (2009) 2319–2339. [CrossRef] [Google Scholar]
  2. G. Allaire and M. Amar, Boundary layer tails in periodic homogenization. ESAIM: COCV 4 (1999) 209–243. [CrossRef] [EDP Sciences] [Google Scholar]
  3. G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization. SIAM Multiscale Model. Simul. 4 (2005) 790–812. [CrossRef] [MathSciNet] [Google Scholar]
  4. A. Anantharaman, Ph.D. thesis, Thèse de l’Université Paris-Est (2011). Available at http://tel.archives-ouvertes.fr/tel-00558618/fr [Google Scholar]
  5. A. Anantharaman, R. Costaouec, C. Le Bris, F. Legoll and F. Thomines, Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments, in Multiscale Modeling and Analysis for Materials Simulation, vol. 22 of Lect. Notes Ser., edited by W. Bao and Q. Du. Institute for Mathematical Sciences, National University of Singapore (2011) 197–272. [Google Scholar]
  6. A. Anantharaman and C. Le Bris, Homogénéisation d’un matériau périodique faiblement perturbé aléatoirement [Homogenization of a weakly randomly perturbed periodic material]. C.R. Acad. Sci. Sér. I 348 (2010) 529–534. [CrossRef] [Google Scholar]
  7. A. Anantharaman and C. Le Bris, A numerical approach related to defect-type theories for some weakly random problems in homogenization. SIAM Multiscale Model. Simul. 9 (2011) 513–544. [CrossRef] [Google Scholar]
  8. A. Anantharaman and C. Le Bris, Elements of mathematical foundations for a numerical approach for weakly random homogenization problems. Commun. Comput. Phys. 11 (2012) 1103-1143. [Google Scholar]
  9. M. Avellaneda and F. H. Lin, Compactness methods in the theory of homogenization. Commun. Pure Appl. Math. 40 (1987) 803–847. [CrossRef] [MathSciNet] [Google Scholar]
  10. G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization. Asymptot. Anal. 59 (2008) 1–26. [MathSciNet] [Google Scholar]
  11. G. Bal and W. Jing, Corrector theory for MsFEM and HMM in random media. SIAM Multiscale Model. Simul. 9 (2011) 1549–1587. [CrossRef] [Google Scholar]
  12. A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, vol. 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, New York (1978). [Google Scholar]
  13. X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization using antithetic variables. Markov Processes and Related Fields 18 (2012) 31–66. (preliminary version available at http://cermics.enpc.fr/˜legoll/hdr/FL24.pdf). [MathSciNet] [Google Scholar]
  14. X. Blanc, C. Le Bris and P.-L. Lions, Une variante de la théorie de l’homogénéisation stochastique des opérateurs elliptiques [A variant of stochastic homogenization theory for elliptic operators]. C.R. Acad. Sci. Sér. I 343 (2006) 717–724. [Google Scholar]
  15. X. Blanc, C. Le Bris and P.-L. Lions, Stochastic homogenization and random lattices. J. Math. Pures Appl. 88 (2007) 34–63. [CrossRef] [Google Scholar]
  16. A. Bourgeat and A. Piatnitski, 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. [MathSciNet] [Google Scholar]
  17. L. Carballal Perdiz, Etude d’une méthodologie multiéchelles appliquée à différents problèmes en milieu continu et discret (in french). Thèse de l’Université Toulouse III (2010). Available at http://thesesups.ups-tlse.fr/1170/. [Google Scholar]
  18. Z. Chen, Multiscale methods for elliptic homogenization problems, Numer. Methods Partial Differ. Eq. 22 (2006) 317–360. [CrossRef] [Google Scholar]
  19. Z. Chen, M. Cui, T. Y. Savchuk and X. Yu, The multiscale finite element method with nonconforming elements for elliptic homogenization problems. SIAM Multiscale Model. Simul. 7 (2008) 517–538. [CrossRef] [Google Scholar]
  20. Z. Chen and T.Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72 (2002) 541–576. [Google Scholar]
  21. Z. Chen and T.Y. Savchuk, Analysis of the multiscale finite element method for nonlinear and random homogenization problems. SIAM J. Numer. Anal. 46 (2008) 260–279. [CrossRef] [Google Scholar]
  22. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland (1978). [Google Scholar]
  23. D. Cioranescu and P. Donato, An introduction to homogenization. In vol. 17 of Oxford Lect. Ser. Math. Appl. The Clarendon Press, Oxford University Press, New York (1999). [Google Scholar]
  24. R. Costaouec, Asymptotic expansion of the homogenized matrix in two weakly stochastic homogenization settings. Appl. Math. Res. Express 2012 (2012) 76–104. [Google Scholar]
  25. R. Costaouec, C. Le Bris and F. Legoll, Approximation numérique d’une classe de problèmes en homogénéisation stochastique [Numerical approximation of a class of problems in stochastic homogenization]. C.R. Acad. Sci. Série I 348 (2010) 99–103. [Google Scholar]
  26. P. Dostert, Y.R. Efendiev and T.Y. Hou, Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification. Comput. Methods Appl. Mechanics Engrg. 197 (2008) 3445–3455. [Google Scholar]
  27. W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. [CrossRef] [MathSciNet] [Google Scholar]
  28. W. E and B. Engquist, The Heterogeneous Multiscale Method for homogenization problems, in Multiscale Methods in Science and Engineering, vol. 44, Lect. Notes Comput. Sci. Engrg. Springer, Berlin (2005) 89–110. [Google Scholar]
  29. W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2 (2007) 367–450. [Google Scholar]
  30. Y.R. Efendiev and T.Y. Hou, Multiscale finite element methods: theory and applications, Surveys and tutorials in the applied mathematical sciences. Springer, New York (2009). [Google Scholar]
  31. Y.R. Efendiev, T.Y. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2 (2004) 553–589. [CrossRef] [Google Scholar]
  32. Y.R. Efendiev, T.Y. Hou and X.-H. Wu, Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal. 37 (2000) 888–910. [CrossRef] [MathSciNet] [Google Scholar]
  33. FreeFEM, http://www.freefem.org [Google Scholar]
  34. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, reprint of the 1998 edn., Classics in Mathematics. Springer (2001). [Google Scholar]
  35. V. Ginting, A. Malqvist and M. Presho, A novel method for solving multiscale elliptic problems with randomly perturbed data. SIAM Multiscale Model. Simul. 8 (2010) 977–996. [CrossRef] [Google Scholar]
  36. A. Gloria, An analytical framework for numerical homogenization. Part II: Windowing and oversampling. SIAM Multiscale Model. Simul. 7 (2008) 274–293. [CrossRef] [Google Scholar]
  37. 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 (1997) 169–189. [CrossRef] [MathSciNet] [Google Scholar]
  38. T.Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913–943. [CrossRef] [MathSciNet] [Google Scholar]
  39. 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. Commun. Math. Sci. 2 (2004) 185–205. [CrossRef] [Google Scholar]
  40. V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994). [Google Scholar]
  41. C. Le Bris, Some numerical approaches for “weakly” random homogenization, Numerical Mathematics and Advanced Applications 2009, in Proc. of ENUMATH 2009. Edited by G. Kreiss et al. Springer Lect. Ser. Notes Comput. Sci. Engrg. (2010) 29–45. [Google Scholar]
  42. C. Le Bris, F. Legoll and F. Thomines, Rate of convergence of a two-scale expansion for some weakly stochastic homogenization problems. Asymptot. Anal. 80 (2012) 237–267. [MathSciNet] [Google Scholar]
  43. F. Legoll and F. Thomines, On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients. ESAIM: M2AN 48 (2014) 347–386. [CrossRef] [EDP Sciences] [Google Scholar]
  44. A. Lozinski, Habilitation à Diriger des Recherches, Université Paul Sabatier, Toulouse (2010). Available at http://www.math.univ-toulouse.fr/. [Google Scholar]
  45. Y. Maday, Reduced basis method for the rapid and reliable solution of partial differential equations, in vol. III of Intern. Congress of Math., Eur. Math. Soc. Zürich (2006) 1255–1270. [Google Scholar]
  46. Y. Mittal, Limiting behavior of maxima in stationary Gaussian sequences. Ann. Probab. 2 (1974) 231–242. [CrossRef] [Google Scholar]
  47. G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in vol. 10 of Proc. Colloq. on Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory, 1979. Edited by J. Fritz, J.L. Lebaritz and D. Szasz. Colloquia Mathematica Societ. J. Bolyai, North-Holland (1981) 835–873. [Google Scholar]
  48. L. Tartar, Estimations of homogenized coefficients, in Topics in the mathematical modelling of composite materials, vol. 31 of Progr. Nonlinear Differ. Equ. Appl., edited by A. Cherkaev and R. Kohn. Birkhäuser (1987). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you