  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.
