Voir la notice de l'article provenant de la source Numdam
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.
@article{M2AN_2014__48_2_347_0,
author = {Legoll, Fr\'ed\'eric and Thomines, Florian},
title = {On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {347--386},
publisher = {EDP-Sciences},
volume = {48},
number = {2},
year = {2014},
doi = {10.1051/m2an/2013111},
mrnumber = {3177849},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013111/}
}
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 PY - 2014 SP - 347 EP - 386 VL - 48 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013111/ DO - 10.1051/m2an/2013111 LA - en ID - M2AN_2014__48_2_347_0 ER -
%0 Journal Article %A Legoll, Frédéric %A Thomines, Florian %T On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 347-386 %V 48 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013111/ %R 10.1051/m2an/2013111 %G en %F M2AN_2014__48_2_347_0
Legoll, Frédéric; 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 , Multiscale problems and techniques. Special Issue, Tome 48 (2014) no. 2, pp. 347-386. doi: 10.1051/m2an/2013111
Cité par Sources :