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A brief introduction to homogenization and miscellaneous applications*

Grégoire Allaire (2012)

ESAIM: Proceedings

This paper is a set of lecture notes for a short introductory course on homogenization. It covers the basic tools of periodic homogenization (two-scale asymptotic expansions, the oscillating test function method and two-scale convergence) and briefly describes the main results of the more general theory of G− or H−convergence. Several applications of the method are given: derivation of Darcy’s law for flows in porous media, derivation of the porosity...

A new numerical scheme for non uniform homogenized problems: application to the nonlinear Reynolds compressible equation.

Buscaglia, Gustavo C., Jai, Mohammed (2001)

Mathematical Problems in Engineering

Alternative approaches to the two-scale convergence

Luděk Nechvátal (2004)

Applications of Mathematics

Two-scale convergence is a special weak convergence used in homogenization theory. Besides the original definition by Nguetseng and Allaire two alternative definitions are introduced and compared. They enable us to weaken requirements on the admissibility of test functions $ψ (x, y)$ . Properties and examples are added.

Analysis of a multiple-porosity model for single-phase flow through naturally fractured porous media.

Spagnuolo, Anna Maria, Wright, Steve (2003)

Journal of Applied Mathematics

Application of homogenization theory related to Stokes flow in porous media

Børre Bang, Dag Lukkassen (1999)

Applications of Mathematics

We consider applications, illustration and concrete numerical treatments of some homogenization results on Stokes flow in porous media. In particular, we compute the global permeability tensor corresponding to an unidirectional array of circular fibers for several volume-fractions. A 3-dimensional problem is also considered.

Application of very weak formulation on homogenization of boundary value problems in porous media

Eduard Marušić-Paloka (2021)

Czechoslovak Mathematical Journal

The goal of this paper is to present a different approach to the homogenization of the Dirichlet boundary value problem in porous medium. Unlike the standard energy method or the method of two-scale convergence, this approach is not based on the weak formulation of the problem but on the very weak formulation. To illustrate the method and its advantages we treat the stationary, incompressible Navier-Stokes system with the non-homogeneous Dirichlet boundary condition in periodic porous medium. The...