Homogenizing the viscoelasticity problem with long-term memory

The system of integro-differential equations describing the small oscillations of an ?-periodic viscoelastic material with long-term memory is considered. Using the two-scale convergencemethod, we construct the systemof homogenized equations and prove the strong convergence as ? → 0 of the solutions of prelimit problems to the solution of the homogenized problem in the norm of the space L ².     

Homogenization of the variational inequality corresponding to a problem with rapidly varying boundary conditions

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on ?-periodically located subsets G _ε belonging to the boundary ?Ω of the domain Ω ? ?³. We construct a limit (homogenized) problem and prove the strong (in H ₁(Ω)) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as ? → 0 in the so-called critical case.     

Homogenization of a non-stationary Stokes flow in porous medium including a layer

We study the behavior of the solution to the non-stationary Stokes equations in a porous medium with characteristic size of the pores ε and containing a thin fissure {0?xn?η} of width η. The limit when ε and η tend to zero gives the homogenized behavior of the flow, which depends on the comparison between ε and η.     

Étude d'un modèle thermo-chimique de formation d'un matériau composite

We consider a composite material constituted of carbon or glass fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period $?>0$. First we prove the existence and uniqueness of a solution by using Schauder's fixed point theorem. Then, by using an asymptotic expansion, we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero and we obtain an error estimate in a case of weak non-linearity. Finally we solve numerically the homogenized problem. To cite this article: S. Meliani et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Convergence theorem for the common solution for a finite family of ?-strongly accretive operator equations

The purpose of this paper is to study a strong convergence of multi-step iterative scheme to a common solution for a finite family of uniformly continuous ?-strongly accretive operator equations in an arbitrary Banach space. As a consequence, the strong convergence theorem for the multi-step iterative sequence to a common fixed point for finite family of ?-strongly pseudocontractive mappings is also obtained. The results presented in this paper thus improve and extend the corresponding results of Inchan [6], Kang [8] and [9] and many others.     

Interior Hölder and gradient estimates for the homogenization of the linear elliptic equations

Hölder and gradient estimates for the correctors in the homogenization are presented based on the translation invariance and Li-Vogelius’s gradient estimate. If the coefficients are piecewise smooth and the homogenized solution is smooth enough, the interior error of the first-order expansion is O(?) in the Hölder norm; it is O(?) in W ^1,∞ based on the Avellaneda-Lin’s gradient estimate when the coefficients are Lipschitz continuous. These estimates can be partly extended to the nonlinear parabolic equations.     

On multiscale homogenization problems in boundary layer theory

This paper is concerned with the homogenization of the equations describing a magnetohydrodynamic boundary layer flow past a flat plate, the flow being subjected to velocities caused by injection and suction. The fluid is assumed incompressible, viscous and electrically conducting with a magnetic field applied transversally to the direction of the flow. The velocities of injection and suction and the applied magnetic field are represented by rapidly oscillating functions according to several scales. We derive the homogenized equations, prove convergence results and establish error estimates in a weighted Sobolev norm and in C ⁰-norm. We also examine the asymptotic behavior of the solutions of the equations governing a boundary layer flow past a rough plate with a locally periodic oscillating structure.     

Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

Polynomial approximations of a class of stochastic multiscale elasticity problems

We consider a class of elasticity equations in \({\mathbb{R}^d}\) whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger–Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli’s expansion, we deduce bounds and summability properties for the solutions’ gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants’ ratio when it goes to \({\infty}\). Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together with the best N term approximation error, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization error that is independent of the ratio of the Lamé constants, so that the error for the corrector is also independent of this ratio.     

Convergence in L space for the homogenization problems of elliptic and parabolic equations in the plane

We study the convergence rate of an asymptotic expansion for the elliptic and parabolic operators with rapidly oscillating coefficients. First we propose homogenized expansions which are convolution forms of Green function and given force term of elliptic equation. Then, using local L^p-theory, the growth rate of the perturbation of Green function is found. From the representation of elliptic solution by Green function, we estimate the convergence rate in L^p space of the homogenized expansions to the exact solution. Finally, we consider L2(0,T:H1(Ω)) or L∞(Ω×(0,T)) convergence rate of the first order approximation for parabolic homogenization problems.     

Newton-type methods of high order and domains of semilocal and global convergence

We present the geometric construction of some classical iterative methods that have global convergence and “infinite” speed of convergence when they are applied to solve certain nonlinear equations f(t)=0. In particular, for nonlinear equations with the degree of logarithmic convexity of f^′, L_f^′(t)=f^′(t)f^?(t)/f^″(t)², is constant, a family of Newton-type iterative methods of high orders of convergence is constructed. We see that this family of iterations includes the classical iterative methods. The convergence of the family is studied in the real line and the complex plane, and domains of semilocal and global convergence are located.     

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N ^?1 lnN) ^p ),p=1,2, on the Shishkin mesh and O(N ^?p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.     

Direct iterative methods for least-squares solutions to singular operator equations

Least-squares consistency and convergence of iterative schemes are investigated for singular operator equations (1) Tx = f, where T is a bounded linear operator from a Banach space to a Hilbert space. A direct splitting of T into T = M ? N is then used to obtain the iterative formula (2) x^(k+1) = M^?Nx^(k) + M^?f, where M^? is a least-squares generalized inverse of M. Cone monotonicity is used to investigate convergence of (2) to a least-squares solutions to (1), extending results given for the matrix case given by Berman and Plemmons.     

Effective boundary condition at a rough surface starting from a slip condition

We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength ?. It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in the limit ε=0 is always no-slip. We give in this paper error estimates for this homogenized no-slip condition, and provide a more accurate effective boundary condition, of Navier type. Our result extends those obtained in Basson and Gérard-Varet (2008) [6] and Gerard-Varet and Masmoudi (2010) [13], in which the special case of a Dirichlet condition at the rough boundary was examined.     

Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties

We study the asymptotic behavior of global classical solutions to hydrodynamical systems modeling the nematic liquid crystal flows under kinematic transports for molecules of different shapes. The coupling system consists of Navier–Stokes equations and kinematic transport equations for the molecular orientations. We prove the convergence of global solutions to single steady states as time tends to infinity as well as estimates on the convergence rate both in 2D for arbitrary regular initial data and in 3D for certain particular cases.     

Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions

The Cauchy problem of the Navier-Stokes-Poisson equations in multi-dimensions (n?3) is considered. We obtain the pointwise estimates of the solution when it is a perturbation of the constant state. Then we have the optimal Lp(Rn) (p?1) convergence rate of the solution.     

Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters 0?<?ε?≤?μ?≤?1, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have boundary layers which overlap and interact, based on the relative size of ε and μ. We show how one can construct full asymptotic expansions together with error bounds that cover the complete range 0?<?ε?≤?μ?≤?1. For the present case of analytic input data, we present derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.     

Diagonalization method for a vector boundary problem of singular perturbation type

The nonlinear boundary value problem ?y″ + f(t, ?, y, y′) = 0, y(0, ?) = α(?), y(1, ?) = β(?), where ? > 0 is a small parameter and y, f are scalar functions, has been studied extensively. However, for n-dimensional vector functions y, f the problem seems open. Here we study this vector boundary problem and obtain results which are analogous to those for the scalar case. The approach in this paper is to transform the appropriate differential equation into a canonical or diagonalized system of two first-order equations.     

Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ?², where ? takes arbitrary values in the interval (0, 1]. When ? vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ? tends to zero, a parabolic boundary layer with a characteristic width ? appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 , where \(N = \mathop {\min }\limits_s N_s \), N ^s + 1 and N ₀ + 1 are the numbers of mesh points on the axes x _s and t, respectively.     

Application of homogenization and large deviations to a parabolic semilinear equation

We study the behavior of the solution of a partial differential equation with a linear parabolic operator with non-constant coefficients varying over length scale δ and nonlinear reaction term of scale 1/?. The behavior is required as ? tends to 0 with δ small compared to ?. We use the theory of backward stochastic differential equations corresponding to the parabolic equation. Since δ decreases faster than ?, we may apply the large deviations principle with homogenized coefficients.