We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the -dimensional lattice with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter)...
We introduce and analyze a numerical strategy
to approximate effective coefficients in stochastic homogenization of discrete elliptic
equations. In particular, we consider the simplest case possible: An elliptic equation on
the -dimensional lattice
with independent and identically distributed conductivities on the associated edges.
Recent results by Otto and the author quantify the error made by approximating
the homogenized coefficient by the averaged energy of a regularized
corrector (with...
We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the
-norm in probability of...
These notes give a state of the art of numerical homogenization methods for linear
elliptic equations. The guideline of these notes is analysis. Most of the numerical
homogenization methods can be seen as (more or less different) discretizations of the same
family of continuous approximate problems, which H-converges to the homogenized problem.
Likewise numerical correctors may also be interpreted as approximations of Tartar’s
correctors. Hence the...
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