Two-scale convergence is a powerful mathematical tool in periodic homogenization developed for modelling media with periodic structure. The contribution deals with the classical definition, its problems, the ``dual'' definition based on the so-called periodic unfolding. Since in the case of domains with boundary the unfolding operator introduced by D. Cioranescu, A. Damlamian, G. Griso does not satisfy the crucial integral preserving property, the contribution proposes a modified unfolding operator...

Reiterated homogenization is studied for divergence structure parabolic problems of the form $\partial u_{\epsilon }/\partial t-div\left(a\left(x,x/\epsilon ,x/{\epsilon }^2,t,t/{\epsilon }^k\right)\nabla u_{\epsilon }\right)=f$. It is shown that under standard assumptions on the function $a(x,y_1,y_2,t,\tau )$ the sequence $\left\{u_{ϵ}\right\}$ of solutions converges weakly in $L^2(0,T;H_0^1\left(\Omega \right))$ to the solution $u$ of the homogenized problem $\partial u/\partial t-div\left(b\right(x,t\left)\nabla u\right)=f$.

This paper is a survey of articles [5, 6, 8, 9, 13, 17, 18]. We are interested in the influence of small geometrical perturbations on the solution of elliptic problems. The cases of a single inclusion or several well-separated inclusions have been deeply studied. We recall here techniques to construct an asymptotic expansion. Then we consider moderately close inclusions, i.e. the distance between the inclusions tends to zero more slowly than their characteristic size. We provide a complete asymptotic...

We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale...

The main purpose of the present paper is to study the asymptotic behavior (when $\epsilon \to 0$) of the solution related to a nonlinear hyperbolic-parabolic problem given in a periodically heterogeneous domain with multiple spatial scales and one temporal scale. Under certain assumptions on the problem’s coefficients and based on a priori estimates and compactness results, we establish homogenization results by using the multiscale convergence method.

In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale ε and use two-scale homogenization techniques to derive effective...

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form $\partial u_{\epsilon }^{\omega }/\partial t+1/{ϵ}_3𝒞\left(T_3(x/{\epsilon }_3){\omega }_3\right)·\nabla u_{\epsilon }^{\omega }-div\left(\alpha \left(T_1(x/{\epsilon }_1){\omega }_1,T_2(x/{\epsilon }_2){\omega }_2,t\right)\nabla u_{\epsilon }^{\omega }\right)=f$. It is shown, under certain structure assumptions on the random vector field $𝒞\left({\omega }_3\right)$ and the random map $\alpha ({\omega }_1,{\omega }_2,t)$, that the sequence $\left\{u_{ϵ}^{\omega }\right\}$ of solutions converges in the sense of G-convergence of parabolic operators to the solution $u$ of the homogenized problem $\partial u/\partial t-div\left(ℬ\right(t\left)\nabla u\right)=f$.

This paper is devoted to the asymptotic behaviour of quadratic forms defined on $L^2$. More precisely we consider the $\mathrm{\Gamma }$-convergence of these functionals for the $L^2$-weak topology. We give an example in which some limit forms are not Markovian and hence the Beurling-Deny representation formula does not hold. This example is obtained by the homogenization of a stratified medium composed of insulating thin-layers.

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 d-dimensional lattice ${\mathbb{Z}}^d$ 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 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 d-dimensional lattice ${\mathbb{Z}}^d$ 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...

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...

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified...

Boundary value problems for the system of linear elasticity with rapidly alternating boundary conditions are studied and asymptotic behavior of solutions is considered when a small parameter, which defines the oscillation of the boundary conditions, tends to zero. Estimates for the difference between such solutions and solutions of the limit problem are given.

We characterize some $G$-limits using two-scale techniques and investigate a method to detect deviations from the arithmetic mean in the obtained $G$-limit provided no periodicity assumptions are involved. We also prove some results on the properties of generalized two-scale convergence.

In this paper the problem of homogeneization for the Laplace operator in partially perforated domains with small cavities and the Neumann boundary conditions on the boundary of cavities is studied. The corresponding spectral problem is also considered.

The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.