Approximations of effective coefficients in stochastic homogenization

Abstract

This note deals with localized approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of "periodization" and other "cut-off" procedures. For instance in the case of periodic approximation, we consider a cubic sample [0, ρ] d of the random medium, extend it periodically in R d and use the effective coefficients of the obtained periodic operators as an approximation of the effective coefficients of the original random operator. It is shown that this approximation converges a.s., as ρ → ∞, and gives back the effective coefficients of the original random operator. Moreover, under additional mixing conditions on the coefficients, the rate of convergence can be estimated by some negative power of ρ which only depends on the dimension, the ellipticity constant and the rate of decay of the mixing coefficients. Similar results are established for approximations in terms of appropriate Dirichlet and Neumann problems localized in a cubic sample [0, ρ] d .

Numerical homogenization: survey, new results, and perspectives

“…In particular, these estimates are expected to be optimal (they have the central limit theorem scaling) up to the logarithmic correction for d = 2. They improve the estimates by Bourgeat and Piatnitski in [12] and by E, Ming, and Zhang in [18], which essentially show (for more general statistics on the coefficients however) that there exists some non explicit exponent γ > 0 such that…”

Tail estimates for homogenization theorems in random media

“…For uniformly elliptic, stationary and ergodic conductances, the effective diffusion matrix of the jump process in a periodic environment converges to the homogenized effective diffusion matrix D 0 (see [5] Th. 1 and [22] Th.…”

“…The approximation of D 0 by periodic environments is considered in [5,6,22]. Given an environment ω ∈ Ω and an integer N > 1, construct an environment N -periodic on In view of applications to the theory of massless gradient fields on Z d , where periodized states can be used to define the slope-independent surface tension [12], Caputo and Ioffe [6] considered periodic approximations of the homogenized diffusion matrix of a symmetric random walk on Z d with i.i.d.…”

“…Bourgeat and Piatnitski [5], considered elliptic differential operators with stationary and uniformly elliptic coefficients. They showed the a.s. convergence of the approximations of the effective diffusion matrix by cubic samples with periodic, Dirichlet or Neumann boundary conditions.…”

Averaging of flows with capillary hysteresis in stochastic porous media

“…A construction of approximate solutions to a one-dimensional unsaturated flow problem can be found in [14]. Homogenization of two-phase flows is performed, for example, in [6,7], a filtration model with hysteresis is studied in [15]. Regarding homogenization of stochastic flow problems, we mention [9,11,13].…”