# On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

Frédéric Legoll; Florian Thomines

- Volume: 48, Issue: 2, page 347-386
- ISSN: 0764-583X

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topLegoll, Frédéric, and Thomines, Florian. "On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 347-386. <http://eudml.org/doc/273324>.

@article{Legoll2014,

abstract = {We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.},

author = {Legoll, Frédéric, Thomines, Florian},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {stochastic homogenization; random stationary diffeomorphisms; central limit result; approximation of homogenized coefficients; central limit theorem},

language = {eng},

number = {2},

pages = {347-386},

publisher = {EDP-Sciences},

title = {On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients},

url = {http://eudml.org/doc/273324},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Legoll, Frédéric

AU - Thomines, Florian

TI - On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 2

SP - 347

EP - 386

AB - We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.

LA - eng

KW - stochastic homogenization; random stationary diffeomorphisms; central limit result; approximation of homogenized coefficients; central limit theorem

UR - http://eudml.org/doc/273324

ER -

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