Homogenization Of Partial Differential Equations Research Articles

Homogenization of partial differential equations with Preisach operators

The current work deals with initial boundary value parabolic problems with Preisach hysteresis whose the density functions are allowed to depend on the variable of space. The model contains nonlinear monotone operators in the diffusion term, arising from an energy. Thanks to the properties of Preisach hysteresis operators and to the sigma-convergence method, we obtain the convergence of themicroscopic solutions to the solution of the homogenized problem. The effective operator is obtained in terms of a solution of a nonlinear corrector equation addressed in the usual sense of distributions, leading in an approximate scheme for the homogenizedcoefficient which is an important step towards the numerical implementation of the results from the homogenization theory beyond the periodic setting.

Cell Averaging Two-Scale Convergence: Applications to Periodic Homogenization

The aim of the paper is to introduce an alternative notion of two-scale convergence which gives a more natural modeling approach to the homogenization of partial differential equations with periodically oscillating coefficients: while removing the bother of the admissibility of test functions, it nevertheless simplifies the proof of all the standard compactness results which made classical two-scale convergence very worthy of interest. Bounded sequences in $L^2_{\sharp} [ {Y}, L^2 (\Omega) ]$ and $L^2_{\sharp} [ {Y}, H^1 (\Omega) ]$ are proven to be relatively compact with respect to this new type of convergence. The strengths of the notion are highlighted on the classical homogenization problem of linear second-order elliptic equations for which first-order boundary corrector-type results are also established. Eventually, possible weaknesses of the method are pointed out on a nonlinear problem: the weak two-scale compactness result for ${\bf S}^2$-valued stationary harmonic maps.

Homogenization of Large-Scale Movement Models in Ecology

A difficulty in using diffusion models to predict large scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). We derive a homogenization procedure for ecological diffusion and apply it to a simple model for chronic wasting disease in mule deer. Homogenization allows us to determine the impact of small scale (10-100 m) habitat variability on large scale (10-100 km) movement. The procedure generates asymptotic equations for solutions on the large scale with parameters defined by small-scale variation. The simplicity of this homogenization procedure is striking when compared to the multi-dimensional homogenization procedure for Fickian diffusion,and the method will be equally straightforward for more complex models.

Deriving Macroscopic Myocardial Conductivities by Homogenization of Microscopic Models

We derive the values for the intracellular and extracellular conductivities needed for bidomain simulations of cardiac electrophysiology using homogenization of partial differential equations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace's equation. Using directly measurable microproperties such as cellular dimensions and end-to-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cyto-architecture (Stinstra et al. in Ann. Biomed. Eng. 33:1743-1751, 2005). This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions along the myocyte membranes.

V.A. Marchenko and E.Y. Khruslov: Homogenization of Partial Differential Equations

This book, which was first published in Russian in 2005, deals with the homogenization of partial differential equations (pdes) of elliptic and parabolic types. The authors study both boundary value problems posed in highly perforated domains or equations with rapidly oscillating coefficients. The standard theory of homogenization (periodic theory, uniformly elliptic oscillating coefficients) is supposed to be already known and the emphasis is put on nonstandard situations leading to multicomponent or nonlocal equations.

Averaging of Backward Stochastic Differential Equations and Homogenization of Partial Differential Equations with Periodic Coefficients

We study the limit of the solutions of systems of semi-linear partial differential equations (PDEs) of second order of parabolic type, with rapidly oscillating periodic coefficients, a singular drift, and singular coefficients of the zero and second order terms. Our basic tool is the approach given by Pardoux [14]. In particular, we use the weak convergence of an associated backward stochastic differential equation (BSDE).

Geometric two-scale convergence on forms and its applications to Maxwell's equations

We develop the geometric two-scale convergence on forms in order to describe the homogenization of partial differential equations with random variables on non-flat domain. We prove the compactness theorem and some two-scale behaviours for differential forms. For its applications, we investigate the limiting equations of the n-dimensional Maxwell equations with random coefficients, with given initial and boundary conditions, where are symmetric positive-definite matrices for x ∈ M, and M is an n-dimensional compact oriented Riemannian manifold with smooth boundary. The limiting system of n-dimensional Maxwell equations turns out to be degenerate and it is proven to be well-posed. The homogenized coefficients affected by the geometry of the domain are presented, and compared with the homogenized coefficient of the second order elliptic equation. We present the convergence theorem in order to explain the convergence of the solutions of Maxwell system as a parabolic partial differential equation.

Homogenization Structures and Applications I

We lay the foundations of a mathematical theory of homogenization structures and show how the latter arises in the homogenization of partial differential equations. We find out that the concept of a homogenization structure turns out to be exactly the right tool that is needed to systematically extend homogenization theory beyond the classical periodic setting. This permits to work out various outstanding nonperiodic homogenization problems that were out of reach till then for lack of an appropriate mathematical framework. The classical Gelfand representation theory is one of our main tools and our basic approach is an adaptation of the two-scale convergence method.

Anisotropic curvature flow in a very thin domain

The asymptotic limit of the anisotropic curvature flow in a very thin domain as its width goes to zero is studied. The problem is formulated by the level set method in the framework of the homogenization of partial differential equations. We show the existence and uniqueness of the limit anisotropic curvature flow in the limit domain, and give rigorously the effective partial differential equation which describes the motion of the flow.

Anisotropic curvature flow in a very thin domain

The asymptotic limit of the anisotropic curvature flow in a very thin domain as its width goes to zero is studied. The problem is formulated by the level set method in the framework of the homogenization of partial differential equations. We show the existence and uniqueness of the limit anisotropic curvature flow in the limit domain, and give rigorously the effective partial differential equation which describes the motion of the flow.

Homogenization and Two-Scale Convergence

Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.