Homogénéisation 2 : Introduction à l'analyse multi-échelle

3rd cycleNous traitons un ensemble assez simple mais pourtant représentatif dans la classe très large des problèmes aux limites avec coefficients rapidement oscillants dans des domaines comportant aussi éventuellement une structure périodique interne (du genre petits trous ou micro-fissures). Ainsi nous nous intéressons à des problèmes scalaires bi-dimensionnels, elliptiques, avec conditions aux limites externes de Dirichlet. En plus de coefficients oscillant de manière périodique avec une période petite devant les proportions du domaine, nous envisagons la présence de cavités internes, distribuées avec la même période, et associées aux conditions aux limites de Dirichlet ou de Neumann. L'intention pour la considération de ces deux possibilités n'est pas le goût de la généralité, mais parce que ces deux cas font apparaître des comportements fondamentalement différents des solutions quand la période devient petite. Nous nous concentrons d'abord sur la structure des solutions quand l'interaction avec le bord externe peut être négligée (situation au demeurant assez artificielle) et mettons en évidence la structure infinie à double échelle (l'une est celle du domaine, l'autre celle de la structure périodique) et les algorithmes qui les relient. Ensuite nous nous attaquons à l'interaction de la structure périodique avec le bord externe du domaine. Nous donnons des réponses dans des situations particulières (il semblerait que le cas général ne soit toujours pas traité d'ailleurs). Nous assistons à l'apparition d'une échelle rapide non périodique près du bord externe, vivant dans la couche limite

Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism

In this paper we prove uniform a priori estimates for transmission problems with constant coefficients on two subdomains, with a special emphasis for the case when the ratio between these coefficients is large. In the most part of the work, the interface between the two subdomains is supposed to be Lipschitz. We first study a scalar transmission problem which is handled through a converging asymptotic series. Then we derive uniform a priori estimates for Maxwell transmission problem set on a domain made up of a dielectric and a highly conducting material. The technique is based on an appropriate decomposition of the electric field, whose gradient part is estimated thanks to the first part. As an application, we develop an argument for the convergence of an asymptotic expansion as the conductivity tends to infinity

Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.Comment: 54 page

Ground Energy of the Magnetic Laplacian in Polyhedral Bodies

The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields and Neumann realization in polyhedral domains is characterized by a hierarchy of model problems. We investigate properties of the model problems (continuity, semi-continuity, existence of generalized eigenfunctions). We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with the help of a classical IMS partition based on adequate coverings of the polyhedral domain, whereas upper bounds are established by a novel construction of quasimodes, qualified as sitting or sliding according to spectral properties of local model problems.Comment: 59 page

On the semiclassical Laplacian with magnetic field having self-intersecting zero set

This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter h > 0 in the case when the magnetic field vanishes along a smooth curve which crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit h $\rightarrow$ 0. We show that each crossing point acts as a potential well, generating a new decay scale of h 3/2 for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in R 2 for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0

Continuity properties of the inf-sup constant for the divergence

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solution of an eigenvalue problem for the Stokes system. This eigenvalue problem does not fall into a class for which standard results about numerical approximations can be applied. Indeed, many reasonable finite element methods do not yield a convergent approximation. In this article, we show that under fairly weak conditions on the approximation of the domain, the LBB constant is an upper semi-continuous shape functional, and we give more restrictive sufficient conditions for its continuity with respect to the domain. For numerical approximations based on variational formulations of the Stokes eigenvalue problem, we also show upper semi-continuity under weak approximation properties, and we give stronger conditions that are sufficient for convergence of the discrete LBB constant towards the continuous LBB constant. Numerical examples show that our conditions are, while not quite optimal, not very far from necessary

Sparse tensor product wavelet approximation of singular functions

International audienceOn product domains, sparse-grid approximation yields optimal, dimension independent convergence rates when the function that is approximated has L^2-bounded mixed derivatives of a sufficiently high order. We show that the solution of Poisson's equation on the n-dimensional hypercube with Dirichlet boundary conditions and smooth right-hand side generally does not satisfy this condition. As suggested by P.-A. Nitsche in [Constr. Approx., 21(1) (2005), pp. 63--81], the regularity conditions can be relaxed to corresponding ones in weighted L^2 spaces when the sparse-grid approach is combined with local refinement of the set of one-dimensional wavelets indices towards the end points. In this paper, we prove that for general smooth right-hand sides, the solution of Poisson's problem satisfies these relaxed regularity conditions in any space dimension. Furthermore, since we remove log-factors from the energy-error estimates from Nitsche's work, we show that in any space dimension, locally refined sparse-grid approximation yields the optimal, dimension independent convergence rate

Koiter Estimate Revisited

We prove a universal energy estimate between the solution of the three-dimensional Lamé system on a thin clamped shell and a displacement reconstructed from the solution of the classical Koiter model. The mid-surface of the shell is an arbitrary smooth manifold with boundary. The bound of our energy estimate only involves the thickness parameter, constants attached to the midsurface, the loading, the two-dimensional energy of the solution of the Koiter model and ''wave-lengths'' associated with this solution. This result is in the same spirit as Koiter's who gave a heuristic estimate in 1970. Taking boundary layers into account, we obtain rigorous estimates, which prove to be sharp in the cases of plates and elliptic shells