An optimal error estimate in stochastic homogenization of discrete elliptic equations

Abstract : This is the second article of a series of papers on stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.~r.~t. the lattice spacing (i.~e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric ''homogenized'' matrix $A_\ho=a_{\ho}\Id$ is characterized by $ \xi\cdot A_{\ho}\xi= \langle(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)\rangle $ for any direction $\xi\in\mathbb{R}^d$, where the random field $\phi$ (the ''corrector'') is the unique solution of $ -\nabla^*\cdot A(\xi+\nabla\phi)\;=\;0 $ in $\Z^d$ such that $\phi(0)=0$, $\nabla \phi$ is stationary and $\expec{\nabla \phi}=0$, $\langle\cdot\rangle$ denoting the ensemble average (or expectation). \medskip In order to approximate the homogenized coefficients $A_\ho$, the corrector problem is usually solved in a box $Q_L=[-L,L)^d$ of size $2L$ with periodic boundary conditions, and the space averaged energy on $Q_L$ defines an approximation $A_L$ of $A_\ho$. Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation $A_L$ converges almost surely to $A_\ho$ as $L \uparrow\infty$. In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size $2L$, but replace the elliptic operator by $ T^{-1}-\nabla^*\cdot A\nabla $ with (typically) $T\sim \sqrt{L}$, as standard in the homogenization literature. We then replace the ensemble average by a space average on $Q_L$, and estimate the overall error on the homogenized coefficients in terms of $L$ and $T$.
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Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2012, 22 (1), pp.1-28. 〈10.1214/10-AAP745〉
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Antoine Gloria, Felix Otto. An optimal error estimate in stochastic homogenization of discrete elliptic equations. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2012, 22 (1), pp.1-28. 〈10.1214/10-AAP745〉. 〈inria-00457020v3〉

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