An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Antoine Gloria; Stefan Neukamm; Felix Otto

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 2, page 325-346
  • ISSN: 0764-583X

Abstract

top
We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green’s function by Marahrens and the third author.

How to cite

top

Gloria, Antoine, Neukamm, Stefan, and Otto, Felix. "An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 325-346. <http://eudml.org/doc/273221>.

@article{Gloria2014,
abstract = {We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green’s function by Marahrens and the third author.},
author = {Gloria, Antoine, Neukamm, Stefan, Otto, Felix},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {stochastic homogenization; homogenization error; quantitative estimate},
language = {eng},
number = {2},
pages = {325-346},
publisher = {EDP-Sciences},
title = {An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations},
url = {http://eudml.org/doc/273221},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Gloria, Antoine
AU - Neukamm, Stefan
AU - Otto, Felix
TI - An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 2
SP - 325
EP - 346
AB - We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green’s function by Marahrens and the third author.
LA - eng
KW - stochastic homogenization; homogenization error; quantitative estimate
UR - http://eudml.org/doc/273221
ER -

References

top
  1. [1] G. Allaire and M. Amar, Boundary layer tails in periodic homogenization. ESAIM: COCV 4 (1999) 209–243. Zbl0922.35014MR1696289
  2. [2] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization. Commun. Pure Appl. Math.40 (1987) 803–847. Zbl0632.35018MR910954
  3. [3] A. Bourgeat and A. Piatnitski, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptotic Anal.21 (1999) 303–315. Zbl0960.60057MR1728027
  4. [4] J.G. Conlon and T. Spencer, Strong convergence to the homogenized limit of elliptic equations with random coefficients. Trans. AMS, in press. Zbl1283.81102
  5. [5] A. Gloria, Fluctuation of solutions to linear elliptic equations with noisy diffusion coefficients. Commun. Partial Differ. Eq.38 (2013) 304–338. Zbl1270.35063MR3009082
  6. [6] A. Gloria, S. Neukamm, and F. Otto, Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. MPI Preprint 91 (2013). Zbl1314.39020MR3302119
  7. [7] A. Gloria, S. Neukamm and F. Otto, Approximation of effective coefficients by periodization in stochastic homogenization. In preparation. Zbl1307.35029
  8. [8] A. Gloria and F. Otto, Quantitative results on the corrector equation in stochastic homogenization of linear elliptic equations. In preparation. Zbl06540728
  9. [9] A. Gloria and F. Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab.39 (2011) 779–856. Zbl1215.35025MR2789576
  10. [10] A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab.22 (2012) 1–28. Zbl06026087MR2932541
  11. [11] R.J. Leveque, Finite difference methods for ordinary and partial differential equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2007). Zbl1127.65080MR2378550
  12. [12] S.M. Kozlov, The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188–202, 327. Zbl0415.60059MR542557
  13. [13] R. Künnemann, The diffusion limit for reversible jump processes on Zd with ergodic random bond conductivities. Commun. Math. Phys.90 (1983) 27–68. Zbl0523.60097MR714611
  14. [14] D. Marahrens and F. Otto, Annealed estimates on the Green’s function. MPI Preprint 69 (2012). Zbl06537679
  15. [15] S.J.N. Mosconi, Discrete regularity for elliptic equations on graphs. CVGMT. Available at http://cvgmt.sns.it/papers/53 (2001). 
  16. [16] A. Naddaf and T. Spencer, Estimates on the variance of some homogenization problems. Preprint (1998). Zbl0871.35010
  17. [17] H. Owhadi, Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Relat. Fields125 (2003) 225–258. Zbl1040.60025MR1961343
  18. [18] G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients. In Random fields, Vol. I, II (Esztergom, 1979), vol. 27 of Colloq. Math. Soc. János Bolyai. North-Holland, Amsterdam (1981) 835–873. Zbl0499.60059MR712714
  19. [19] V.V. Yurinskiĭ, Averaging of symmetric diffusion in random medium. Sibirskii Matematicheskii Zhurnal27 (1986) 167–180. Zbl0614.60051MR867870

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.