Partant du principe de conservation de la masse et du principefondamental de la dynamique, on retrouvel'équation d'Euler nous permettant de décrire les modèlesasymptotiques de propagation d'ondes dans des eaux peu profondesen dimension 1. Pour décrire la propagation des ondes en dimension2, Kadomtsev et Petviashvili [ 15 (1970) 539] utilisent une perturbationlinéaire de l'équation de KdV. Mais cela ne précise pas si leséquations ainsi obtenues dérivent de l'équation d'Euler, c'est ceque montrent Ablowitz et Segur dans l'article [J. Fluid Mech.92 (1979) 691–715]. Oninsistera, de la même manière, sur le fait que les équations deKP-BBM peuvent être aussi obtenues à partir de l'équation d'Euler,et dans quelle mesure elles décrivent le modèle physique. Dans unsecond temps, on reprend la méthode introduite dans l'article deBona et al. [Lect. Appl. Math.20 (1983) 235–267] dans lequel ils comparent lessolutions d'ondes longues en dimension 1, à savoir les solutionsdes équations KdV et BBM, pour montrer ici que les solutions deséquations KP-II et KP-BBM-II sont proches sur un intervalle de tempsinversement proportionnel à l'amplitude des ondes. Du point de vue de la modélisation,il sera clair, d'après la première partie, que seul le modèledécrit par KP-BBM-II est bien posé, et comme du point de vuephysique, KP-II et KP-BBM-II décrivent les ondes longues de faibleamplitude lorsque la tension de surface est négligeable, il est intéressant de les comparer.De plus, on verra que la méthode utilisée ici reste valable pourles problèmes périodiques.

This paper is concerned with the unilateral contact problem in linear elasticity. We define two a posteriori error estimators of residual type to evaluate the accuracy of the mixed finite element approximation of the contact problem.Upper and lower bounds of the discretization error are proved forboth estimators and several computations are performed toillustrate the theoretical results.

We introduce and analyse a mixed formulation of theMonge-Kantorovich equations, which express optimality conditions forthe mass transportation problem with cost proportional to distance.Furthermore, we introduce and analyse the finite elementapproximation of this formulation using the lowest orderRaviart-Thomas element. Finally, we present some numericalexperiments, where both the optimal transport density and theassociated Kantorovich potential are computed for a coupling problemand problems involving obstacles and regions of cheaptransportation.


The ground-state energy and properties of any many-electron atom ormolecule may be rigorously computed by variationally computing thetwo-electron reduced density matrix rather than the many-electronwavefunction. While early attempts fifty years ago to compute theground-state 2-RDM directly were stymied because the 2-RDM must beconstrained to represent an N-electron wavefunction, recentadvances in theory and optimization have made direct computation ofthe 2-RDM possible. The constraints in the variational calculationof the 2-RDM require a special optimization known as a semidefiniteprogramming. Development of first-order semidefinite programmingfor the 2-RDM method has reduced the computational costs of thecalculation by orders of magnitude [Mazziotti, Phys. Rev. Lett.93 (2004) 213001]. The variational 2-RDM approach is effective atcapturing multi-reference correlation effects that are especiallyimportant at non-equilibrium molecular geometries. Recent work on2-RDM methods will be reviewed and illustrated with particularemphasis on the importance of advances in large-scale semidefiniteprogramming.


We present a novel application of best N-term approximation theoryin the framework of electronic structure calculations. The paper focusses on thedescription of electron correlations within a Jastrow-type ansatz for thewavefunction. As a starting point we discuss certain natural assumptions onthe asymptotic behaviour of two-particle correlation functions $\mathcal{F}^{(2)}$ near electron-electron and electron-nuclear cusps. Basedon Nitsche's characterization of best N-term approximation spaces $A_{q}^{\alpha}(H^{1})$ , we prove that $\left.\mathcal{F}^{(2)}\inA_{q}^{\alpha}(H^{1})\right.$ for q>1 and $\alpha=\frac{1}{q}-\frac{1}{2}$ with respect to a certain class of anisotropic wavelet tensor product bases.Computational arguments are given in favour of this specific class compared toother possible tensor product bases. Finally, we compare the approximationproperties of wavelet bases with standard Gaussian-type basis sets frequentlyused in quantum chemistry.


This paper deals with the mortar spectral element discretization of two equivalent problems, the Laplace equation and the Darcy system, in a domain which corresponds to a nonhomogeneous anisotropic medium. The numerical analysis of the discretization leads to optimal error estimates and the numerical experiments that we present enable us to verify its efficiency.

Based on estimates for the KdV equation in analyticGevrey classes, a spectral collocation approximation ofthe KdV equation is proved to converge exponentially fast.

This paper is concerned with optimal design problems with aspecial assumption on the coefficients of the state equation.Namely we assume that the variations of these coefficientshave a small amplitude. Then, making an asymptotic expansionup to second order with respect to the aspect ratio of thecoefficients allows us to greatly simplify the optimal designproblem. By using the notion of H-measures we are able toprove general existence theorems for small amplitudeoptimal design and to provide simple and efficient numericalalgorithms for their computation. A key feature of thistype of problems is that the optimal microstructures arealways simple laminates.

In this paper we propose an original approach for the simulation of the time-dependent response of a floating elastic plate using the so-called Singularity Expansion Method. This method consists in computing an asymptotic behaviour for large time obtained by means of the Laplace transform by using the analytic continuation of the resolvent of the problem. This leads to represent the solution as the sum of a discrete superposition of exponentially damped oscillating motions associated to the poles of the analytic continuation called resonances of the system, and a low frequency component associated to a branch point at frequency zero.We present the mathematical analysis of this method for the two-dimensional sea-keeping problem of a thin elastic plate (ice floe, floating runway, ...) and provide some numerical results to illustrate and discuss its efficiency.

In this paper, we compare a biomechanics empirical model of the heart fibrous structure to two models obtained by a non-periodic homogenization process. To this end, the two homogenized models are simplified using the small amplitude homogenization procedure of Tartar, both in conduction and in elasticity. A new small amplitude homogenization expansion formula for a mixture of anisotropic elastic materials is also derived and allows us to obtain a third simplified model.

Motivated by well-driven flow transport in porous media, Chenand Yue proposed a numerical homogenization method for Greenfunction [Multiscale Model. Simul.1 (2003) 260–303]. In that paper,the authors focused on the well pore pressure, so the local erroranalysis in maximum norm was presented. As a continuation, we willconsider a fully discrete scheme and its multiscale error analysis on the velocity field.

This paper reviews popular acceleration techniques to converge the non-linear self-consistent field equations appearing in quantum chemistry calculations with localized basis sets. The different methodologies, as well as their advantages and limitations are discussed within the same framework. Several illustrative examples of calculations are presented. This paper attempts to describe recent achievements and remaining challenges in this field.

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameterdependence to problems involving (a) nonaffine dependence on theparameter, and (b) nonlinear dependence on the field variable.The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computationaldecomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateralreduced-basis approximation space, and (ii) a stable and inexpensiveinterpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in eachinstance, we discuss the reduced-basis approximation and the associated offline-online computationalprocedures. Numerical results are presented to assess our approach.


Systems of mixed hyperbolic-elliptic conservation laws can serve as models for the evolution of a liquid-vapor fluid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost fluid type to solve the sharp-interface model numerically. In particular the approach allows to resolve phase transitions sharply, i.e., without artificial smearing in the physically irrelevant elliptic region. Numerical experiments demonstrate the reliability of the method.

In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uE of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uE based on a multiscale superposition of the unperturbed solution u 0 and a profile defined in a model domain. We conclude with numerical results.

In a foregoing paper [Sonar, ESAIM: M2AN39 (2005) 883–908] we analyzed the Interpolating Moving Least Squares (IMLS) method due to Lancaster and Šalkauskas with respect to its approximation powers and derived finite difference expressions for the derivatives. In this sequel we follow a completely different approach to the IMLS method given by Kunle [Dissertation (2001)]. As a typical problem with IMLSmethod we address the question of getting admissible results at the boundary by introducing “ghost points”.Most interesting in IMLS methods are the finite difference operators whichcan be computed from different choices of basis functions and weightfunctions. We present a way of deriving these discrete operators inthe spatially one-dimensional case. Multidimensional operatorscan be constructed by simply extending our approach to higher dimensions.Numerical results ranging from 1-d interpolation to the solution of PDEs are given.

We explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applicationsthe dynamics near a conical intersection of potential surfaces and reactive scattering.

The present paper proposes and analyzes a general locking free mixed strategy for computing the deformation of incompressible three dimensional structures placed inside flexible membranes. The model involves as in Chapelle and Ferent [Math. Models Methods Appl. Sci.13 (2003) 573–595] a bending dominated shell envelope and a quasi incompressible elastic body. The present work extends an earlier work ofArnold and Brezzi [Math Comp.66 (1997) 1–14] treating the shell part and proposes a global stable finite element approximation by coupling optimal mixed finite element formulations of the different subproblems by mortar techniques. Examples of adequate finite elements are proposed. Convergence results are derived in two steps. First a global inf-sup condition is proved, deducedfrom the local conditions to be satisfied by the finite elements used for the external shell problem, the internal incompressible 3D problem, and the mortar coupling, respectively. Second, the analysis ofArnold and Brezzi [Math. Comp.66 (1997) 1–14] is extended to the present problem and least to convergence results for the full coupled problem, withconstants independent of the problem's small parameters.

The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classicalvariational approach without and with a regularization term in order tosmooth the contours of the classified image. Then we present the generaltopological asymptotic analysis, and we finally introduce its application tothe grey-level image classification problem.

The primary objective of this work is to develop coarse-grainingschemes for stochastic many-body microscopic models and quantify theireffectiveness in terms of a priori and a posteriori error analysis. Inthis paper we focus on stochastic lattice systems ofinteracting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grainedapproximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first – and often inadequate – approximation and obtain more accurate coarse-graining schemes.The cluster expansions yield also sharp a posteriori error estimates forthe coarse-grained approximations that can be used for the construction ofadaptive coarse-graining methods.
We present a number of numerical examples that demonstrate that thecoarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improvepredictions of earlier coarse-graining schemes for short-range interactions.