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.

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 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 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 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.

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.

We study an atomistic pair potential-energy E (n)(y) that describesthe elastic behavior of two-dimensional crystals with n atoms where $y \in {\mathbb R}^{2\times n}$ characterizes the particle positions. The mainfocus is the asymptotic analysis of the ground state energy as ntends to infinity. We show in a suitable scaling regime where theenergy is essentially quadratic that the energy minimum of E (n)admits an asymptotic expansion involving fractional powers of n:

${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$

The bulk energy density E bulk is given by an explicitexpression involving the interaction potentials. The surface energyE surface can be expressed as a surface integral where theintegrand depends only on the surface normal and the interactionpotentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggestthat the integrand is a continuous, but nowhere differentiable function ofthe surface normal.

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 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.

For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.

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.



The Diffusion Monte Carlo method is devoted to the computation ofelectronic ground-state energies of molecules. In this paper, we focus onimplementations of this method which consist in exploring theconfiguration space with a fixed number of random walkers evolvingaccording to a stochastic differential equation discretized in time. Weallow stochastic reconfigurations of the walkers to reduce thediscrepancy between the weights that they carry. On a simpleone-dimensional example, we prove the convergence of the method for afixed number of reconfigurations when the number of walkers tends to+∞ while the timestep tends to 0. We confirm our theoreticalrates of convergence by numerical experiments. Various resamplingalgorithms are investigated, both theoretically and numerically.


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.

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.

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressiblefluid exhibiting phase transitions between a liquid and a vapor phase in the presence ofcapillarity effects close to phase boundaries. Standard numerical discretizations areknown to violate discrete versions of inherent energy inequalities, thus leading tospurious dynamics of computed solutions close to static equilibria (e.g.,parasitic currents). In this work, we propose a time-implicit discretization of theproblem, and use piecewise linear (or bilinear), globally continuous finite element spacesfor both, velocity and density fields, and two regularizing terms where correspondingparameters tend to zero as the mesh-size h > 0 tends to zero.Solvability, non-negativity of computed densities, as well as conservation of mass, and adiscrete energy law to control dynamics are shown. Computational experiments are providedto study interesting regimes of coefficients for viscosity and capillarity.

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete ellipticequations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice $\mathbb{Z}^d$with independent and identically distributed conductivities on the associated edges.Recent results by Otto and the author quantify the error made by approximatingthe homogenized coefficient by the averaged energy of a regularizedcorrector (with parameter T) on some box of finite size L. In this article, we replace the regularizedcorrector (which is the solution of a problem posed on $\mathbb{Z}^d$) by some practically computable proxy on some box of size R ≥ L, and quantify the associated additional error.In order to improve the convergence, one may also consider N independentrealizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients.A natural optimization problem consists in properly choosing T, R, L and N in order toreduce the error at given computational complexity.Our analysis is sharp and sheds some light on this question.In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errorswe derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B modelwithout convective terms is considered. Assuming the domain to be a convexpolygon, existence of a solutionis proved for small relaxation times.Continuous piecewise linear finite elements together witha Galerkin Least Square (GLS) method are studied for solving this problem.Existence and a priori error estimatesare established using a Newton-chord fixed point theorem,a posteriori error estimates are also derived.An Elastic Viscous Split Stress (EVSS) scheme related to the GLS methodis introduced. Numerical results confirm the theoretical predictions.

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.

A new approximation scheme is presented for the mathematical model of convection-diffusion and adsorption. The method is based on the relaxation method and the method of characteristics. We prove the convergence of the method and present some numerical experiments in 1D. The results can be applied to the model of contaminant transport in porous media with multi-site, equilibrium and non-equilibrium type of adsorption.