Monique Dauge

Martin Costabel, Monique Dauge

The equivalence between the inequalities of Babuška-Aziz and Friedrichs for sufficiently smooth bounded domains in the plane has been shown by Horgan and Payne 30 years ago. We prove that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. For the Horgan-Payne inequality, which is an upper bound of the Friedrichs constant for plane star-shaped domains in terms of a geometric quantity known as the Horgan-Payne angle, we show that it is true for some classes of domains, but not for all bounded star-shaped domains. We prove a weaker inequality that is true in all cases.

Monique Dauge, Yvon Lafranche, Nicolas Raymond

The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend essentially on a sole parameter, the opening of the V. The free energy band is a semi-infinite interval bounded from below. As soon as the V is not flat, there are bound states below the free energy band. There are a finite number of them, depending on the opening. This number tends to infinity as the opening tends to 0 (sharply bent V). In this situation, the eigenfunctions concentrate and become self-similar. In contrast, when the opening gets large (almost flat V), the eigenfunctions spread and enjoy a different self-similar structure. We explain all these facts and illustrate them by numerical simulations.

Christine Bernardi, Martin Costabel, Monique Dauge et al.

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.

Monique Dauge, Benjamin Texier

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.

Marie Chaussade-Beaudouin, Monique Dauge, Erwan Faou et al.

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$ε$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $λ$($ε$) and the eigenvector angular frequency k($ε$) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts. First, the scalar Laplace operator for acoustics is addressed, for which k($ε$) is always zero. In contrast to it, for the Lam{é} system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k($ε$) tends to infinity as $ε$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $λ$($ε$) and k($ε$) and demonstrate by numerical examples the different behavior as $ε$ tends to zero.

Marie Chaussade-Beaudouin, Monique Dauge, Erwan Faou et al.

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{é} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{é} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{é} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=γ\varepsilon^{-β}$ with $β=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $β$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented.

Monique Dauge, Jean-Philippe Miqueu, Nicolas Raymond

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.

Samuel Shannon, Zohar Yosibash, Monique Dauge et al.

This report presents explicit analytical expressions for the primal, primal shadows, dual and dual shadows functions for the Laplace equation in the vicinity of a circular singular edge with Neumann boundary conditions on the faces that intersect at the singular edge. Two configurations are investigated: a penny-shaped crack and a 90^o V-notch.

Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond

The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit. On the one hand, we prove that, for any aperture, the eigenvalues accumulate below the thresh-old of the essential spectrum: For a small distance from the essential spectrum, the number of eigenvalues farther from the threshold than this distance behaves like the logarithm of the distance. On the other hand, in the small aperture regime, we provide a two-term asymptotics of the first eigenvalues thanks to a priori localization estimates for the associated eigenfunctions. We prove that these eigenfunctions are localized in the conical cap at a scale of order the cubic root of the aperture angle and that they get into the other part of the layer at a scale involving the logarithm of the aperture angle.

Martin Costabel, Monique Dauge, Serge Nicaise

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.

Daniele Boffi, Martin Costabel, Monique Dauge et al.

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincaré lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, Nédélec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.

Gabriel Caloz, Monique Dauge, Victor Péron

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.