The numerical solution of the Hartree-Fock equations is a central problem in quantumchemistry for which numerous algorithms exist. Attempts to justify these algorithmsmathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod.Numer. Anal. 34 (2000) 749–774], but, to our knowledge, nocomplete convergence proof has been published, except for the large-Zresult of [M. Griesemer and F. Hantsch, Arch. Rational Mech. Anal. (2011)170]. In this paper, we prove the convergence of a natural gradient algorithm, using agradient inequality for analytic functionals due to Łojasiewicz [Ensemblessemi-analytiques. Institut des Hautes Études Scientifiques (1965)]. Then,expanding upon the analysis of [E. Cancès and C. Le Bris, Math. Mod. Numer. Anal.34 (2000) 749–774], we prove convergence results for the Roothaanand Level-Shifting algorithms. In each case, our method of proof provides estimates on theconvergence rate. We compare these with numerical results for the algorithms studied.

In this paper we present a methodology for constructing accurateand efficient hybrid central-upwind (HCU) type schemes forthe numerical resolution of a two-fluid model commonly used by thenuclear and petroleum industry. Particularly, we propose a methodwhich does not make use of any information about theeigenstructure of the Jacobian matrix of the model. The two-fluid model possesses a highly nonlinear pressure law. From the mass conservation equations we develop an evolutionequation which describes how pressure evolves in time. By applyinga quasi-staggered Lax-Friedrichs type discretization for thispressure equation together with a Modified Lax-Friedrich typediscretization of the convective terms, we obtain a central typescheme which allows to cope with the nonlinearity (nonlinearpressure waves) of the two-fluid model in a robust manner.Then, in order to obtain an accurate resolution of mass fronts, weemploy a modification of the convective mass fluxes by hybridizingthe central type mass flux components with upwind type components.This hybridization is based on a splitting of the mass fluxes intocomponents corresponding to the pressure and volume fractionvariables, recovering an accurate resolution of a contactdiscontinuity. In the numerical simulations, the resulting HCU scheme givesresults comparable to an approximate Riemann solver while beingsuperior in efficiency. Furthermore, the HCU scheme yields betterrobustness than other popular Riemann-free upwind schemes.

We show that Boltzmann's collision operator can be written explicitlyin divergence and double divergence forms. These conservativeformulations may be of interest for both theoretical and numericalpurposes. We give an application to the asymptotics of grazing collisions.

We propose a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cutoff hypothesis, and for intermolecular laws behaving as 1/rs , with s> 2. It involves natural pseudo differential operators, under a form which is analogous to the Landau operator. It may be used in the study of the associated equations, and more precisely in the non homogeneous framework.

We examine the effect of numerical integration on the accuracy of high order conforming pyramidal finite element methods. Non-smooth shape functions are indispensable to the construction of pyramidal elements, and this means the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include non-smooth functions and show that, despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.

This paper is concerned with the finite volume approximation of the p-Laplacian equation with homogeneous Dirichlet boundary conditions on rectangular meshes.A reconstruction of the norm of the gradient on the mesh's interfaces is needed in order to discretize the p-Laplacian operator. We give a detailed description of the possible nine points schemes ensuring that the solution of the resulting finite dimensional nonlinear system exists and is unique. These schemes, called admissible, are locally conservative and in addition derive from the minimization of a strictly convexe and coercive discrete functional.The convergence rate is analyzed when the solution lies in W2,p . Numerical results are given in order to compare different admissible and non-admissible schemes.

In this work we describe an efficient model for the simulation of atwo-phase flow made of a gas and a granular solid. The starting point is the two-velocitytwo-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented bya relaxation source term in orderto take into account the pressure equilibrium between the two phases andthe granular stress in the solid phase. We show that the relaxationprocess can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on a splitting approach. Each step of the time marchingalgorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fractionis updated in order to take into account the equilibrium source term.The whole procedure is entropy dissipative.For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volumefraction stays within its natural bounds.

We study the linearized water-wave problem in a bounded domain (e.g. afinite pond of water) of ${\mathbb R}^3$, having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered thatin this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point ${\mathcal O}$ of the water surface, wherea submerged body touches the surface (see Fig. 1). The radiation conditions emerge from the requirement thatthe linear operator associated to the problem be Fredholm of index zeroin relevant weighted function spaces with separated asymptotics.The classification of incoming and outgoing (seen from ${\mathcal O}$) waves and the unitary scattering matrix are introduced.

Different effective boundary conditions or wall laws for unsteady incompressible Navier-Stokes equations over rough domains are derived in the laminar setting. First and second order unsteady wall laws are proposed using two scale asymptotic expansion techniques. The roughness elements are supposed to be periodic and the influence of the rough boundary is incorporated through constitutive constants. These constants are obtained by solving steady Stokes problems and so they are calculated only once. Numerical tests are presented to validate and compare the proposed boundary conditions.

We show that the Maxwell equations in the low frequency limit, in a domain composed of insulatingand conducting regions, has a saddle point structure, wherethe electric field in the insulating region is the Lagrangemultiplier that enforces the curl-free constraint on the magnetic field.We propose a mixed finite element techniquefor solving this problem, and we show that, under mild regularityassumption on the data, Lagrange finite elements can be usedas an alternative to edge elements.

A linearly convergent iterative algorithm that approximates the rank-1 convex envelope $f^{rc}$ of a given function $f:\mathbb{R}^{n\times m} \to \mathbb{R}$, i.e. the largest function below f which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.

A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element method of Raviart–Thomas is proved to converge with a quasi-optimal error bound.

The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss the general low Mach number limit for standard compressible flows given in P.–L. Lions' book that means with constant viscosity coefficients.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methodsfor the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$ , we obtain error estimates in L 2 of order $\mathcal{O} (\Delta t^2 + h^{m + 1/2})$ where m is the degree of the local polynomials.

The aim of this paper is to present a method using both the ideas of sectionalapproach and moment methods in order to accurately simulate evaporationphenomena in gas-droplets flows. Using the underlying kinetic interpretation ofthe sectional method [Y. Tambour, Combust. Flame60 (1985)15–28] exposed in [F. Laurent and M. Massot, Combust. Theory Model.5 (2001) 537–572], we propose an extension of thisapproach based on a more accurate representation of the droplet size numberdensity in each section ensuring the exact conservation of two moments (asopposed to only one moment used in the classical approach). A correspondingsecond-order numerical scheme, with respect to space and droplet size variables,is also introduced and can be proved to be positive and to satisfy a maximumprinciple on the velocity and the mean droplet mass under a suitable CFL-likecondition. Numerical simulations have been performed and the results confirm theaccuracy of this new method even when a very coarse mesh for the droplet sizevariable (i.e.: a low number of sections) is used.

We address multiscale elliptic problems with random coefficients that are a perturbationof multiscale deterministic problems. Our approach consists in taking benefit of theperturbative context to suitably modify the classical Finite Element basis into adeterministic multiscale Finite Element basis. The latter essentially shares the sameapproximation properties as a multiscale Finite Element basis directly generated on therandom problem. The specific reference method that we use is the Multiscale Finite ElementMethod. Using numerical experiments, we demonstrate the efficiency of our approach and thecomputational speed-up with respect to a more standard approach. In the stationarysetting, we provide a complete analysis of the approach, extending that available for thedeterministic periodic setting.

We prove an a priori error estimate for the hp-version of the boundaryelement method with hypersingular operators on piecewise plane open orclosed surfaces. The underlying meshes are supposed to be quasi-uniform.The solutions of problems on polyhedral or piecewise plane open surfaces exhibittypical singularities which limit the convergence rate of the boundary element method.On closed surfaces, and for sufficiently smooth given data, the solution isH 1-regular whereas, on open surfaces, edge singularities arestrong enough to prevent the solution from being in H 1.In this paper we cover both cases and, in particular, prove an a priorierror estimate for the h-version with quasi-uniform meshes.For open surfaces we prove a convergence like O(h1/2p-1),h being the mesh size and p denoting the polynomial degree.This result had been conjectured previously.

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.


We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data. Sufficient conditions on the law of the initial data for moments of the random entropy solution to be piece-wise smooth functions of space and time are identified, even in cases where path-wise random entropy solutions are discontinuous almost surely. We extrapolate the results to hyperbolic systems of conservation laws in one space dimension. We then exhibit a class of stochastic Galerkin discretizations which allows to derive closed deterministic systems of hyperbolic conservation laws for the coefficients in truncated polynomial chaos expansions of the random entropy solution. Based on the regularity theory developed here, we show that depending on the smoothness of the law of the initial data, arbitrarily high convergence rates are possible for the computation of coefficients in gPC approximations of random entropy solutions for Riemann problems with random shock location by combined Stochastic Galerkin Finite Volume schemes.

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equationin $\mathbb{R}^d$ , d=2 or 3,using backward Euler's scheme. For this discretization, we derive a residual indicator, which usea spatial residual indicator based on thejumps of normal and tangential derivatives of the nonconformingapproximation and a time residual indicator based on the jump of broken gradients at each time step.Lower and upper bounds form the main results with minimal assumptions on the mesh.Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.