179 research outputs found
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
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
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
Problèmes de transmission non coercifs dans des polygones
This paper was written in 1997 and published as IRMAR report number 97-27. This work is put on open archive in 2011 in view of recent applications to meta-materials.We show that Kondratev's theory for corner domains can be extended to scalar transmission problems between two polygonal materials if the contrast ratio between the two materials is different from -1
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