Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields

Abstract

A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients requires solving local corrector problems followed by upscaling relevant local data. The most naive way of computing homogenized coefficients is by solving a local elliptic problem, which is known to suffer from the so-called resonance error dominating all other errors inherent in multiscale computations. A far more efficient modelling strategy, based on adding an exponential correction term to the standard local elliptic problem, has recently been proved to result in exponentially decaying error bounds with respect to the size of the local geometry. The questions in relation with the accuracy and computational efficiency of this approach has been previously addressed in the context of periodic homogenization. The present article concerns the extension of mathematical and numerical study of this modified elliptic corrector problem to stochastic homogenization problems. In particular, we assume a stationary, ergodic micro-structure and i) establish the well-posedness of the corrector equation, ii) analyse the bias (or the systematic error) originating from additional exponential correction term in the model. Numerical results corroborating our theoretical findings are presented.

Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields

“…We finally state the main result of this section, which is the sub-optimal moment bound on the time dependent flux q r (T, •) for 1 2 ≤ r ≤ √ T .…”

“…(ii) Second, the semigroup u e has been used more recently in [1] for approximating φ e via exponential regularization, that is we replace the corrector equation (1.1) by u e (T ) − ∇ • a(∇φ e,T,R + e) = 0 in Q R , φ e,T,R ≡ 0 on ∂Q R , for R ≫ 1 and T ≫ 1. Optimal estimates on u e are used to control the bias (or the systematic error).…”

Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields

“…Without loss of generality, we may assume that R ≥ 2r * (0). Indeed, otherwise, we deduce from the 1 8 -Lipschitz property of r * (x) in form of…”

“…We refer to [16] for an analysis via a semigroup approach. (ii) Second, the semigroup u e has been used more recently in [1] for approximating φ e via exponential regularization, that is we replace the corrector equation (1.1) by u e (T ) − ∇ • a(∇φ e,T ,R + e) = 0 in Q R , φ e,T ,R ≡ 0 o n ∂Q R ,…”