Asymptotic Expansion for Multiscale Problems on Non-periodic Stochastic Geometries

TL;DR

This paper extends the asymptotic expansion method from periodic to non-periodic stochastic geometries, showing that results from periodic cases are applicable to certain stochastic structures, with applications in diffusion and porous media flow.

Contribution

It generalizes the asymptotic expansion technique to stationary ergodic stochastic geometries, bridging the gap between periodic and non-periodic structures in homogenization.

Findings

Asymptotic expansion results apply to non-periodic stochastic geometries.

Method successfully applied to diffusion and porous media flow.

Examples demonstrate applicability to various stochastic geometries.

Abstract

The asymptotic expansion method is generalized from the periodic setting to stationary ergodic stochastic geometries. This will demonstrate that results from periodic asymptotic expansion also apply to non-periodic structures of a certain class. In particular, the article adresses non-mathematicians who are familiar with asymptotic expansion and aims at introducing them to stochastic homogenization in a simple way. The basic ideas of the generalization can be formulated in simple terms, which is basically due to recent advances in mathematical stochastic homogenization. After a short and formal introduction of stochastic geometry, calculations in the stochastic case will be formulated in a way that they will not look different from the periodic setting. To demonstrate that, the method will be applied to diffusion with and without microscopic nonlinear boundary conditions and to porous media flow. Some examples of stochastic geometries will be given.