This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis -which has been made possible by recent contributions on quantitative stochastic homogenization theory by two of us and by Otto. This article makes the connection between our theoretical results and computations. We give a complete picture of the numerical methods found in the literature, compare them in terms of known (or expected) convergence rates, and study them numerically. Two types of methods are presented: methods based on the corrector equation, and methods based on random walks in random environments. The numerical study confirms the sharpness of the analysis (which it completes by making precise the prefactors, next to the convergence rates), supports some of our conjectures, and calls for new theoretical developments.

In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim [12] and under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed.

In the first part of this paper, we prove Hölder and logarithmic stability estimates associated with the unique continuation property for the Stokes system. The proof of these results is based on local Carleman inequalities. In the second part, these estimates on the fluid velocity and on the fluid pressure are applied to solve an inverse problem: we consider the Stokes system completed with mixed Neumann and Robin boundary conditions, and we want to recover the Robin coefficient (and obtain the stability estimate for it) from measurements available on a part of the boundary where the Neumann conditions are prescribed. For this identification parameter problem, we obtain a logarithmic stability estimate under the assumption that the velocity of a given reference solution stays far from zero on a part of the boundary where the Robin conditions are prescribed.

We are interested in the inverse problem of recovering a Robin coefficient defined on some non accessible part of the boundary from available data on another part of the boundary in the nonstationary Stokes system. We prove a Lipschitz stability estimate under the a priori assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework. Résumé Estimation de stabilité Lipschitzienne pour le système de Stokes avec des conditions aux limites de types RobinNous nous intéressonsà l'identification d'un coefficient de Robin défini sur une partie non accessible du bordà partir de mesures disponibles sur une autre partie du bord dans le système de Stokes non stationnaire. Nous prouvons une estimation de stabilité Lipschitzienne sous l'hypothèse a priori que le coefficient de Robin est défini dans un sous-ensemble compact et convexe d'un sous-espace vectoriel de dimension finie de l'espace des fonctions continues. Pour ce faire, nous utilisons un théorème prouvé par L. Bourgeois permettant d'établir des inégalités de stabilité Lipschitzienne pour une classe de problèmes inverses dans un cadre abstrait.