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

Antoine Gloria; Stefan Neukamm; Felix Otto

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

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topGloria, 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

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