This study deals with the existence and uniqueness of solutions to dynamical problems of finite freedom involving unilateral contact and Coulomb friction. In the frictionless case, it has been established [P. Ballard, Arch. Rational Mech. Anal. 154 (2000) 199–274] that the existence and uniqueness of a solution to the Cauchy problem can be proved under the assumption that the data are analytic, but not if they are assumed to be only of class C∞. Some years ago, this finding was extended [P. Ballard and S. Basseville, Math. Model. Numer. Anal. 39 (2005) 59–77] to the case where Coulomb friction is included in a model problem involving a single point particle. In the present paper, the existence and uniqueness of a solution to the Cauchy problem is proved in the case of a finite collection of particles in (possibly non-linear) interactions.

In this paper we develop and study numerically a model to describe some aspects of soundpropagation in the human lung, considered as a deformable and viscoelastic porous medium(the parenchyma) with millions of alveoli filled with air. Transmission of sound throughthe lung above 1 kHz is known to be highly frequency-dependent. We pursue the key ideathat the viscoelastic parenchyma structure is highly heterogeneous on the small scaleε and use two-scale homogenization techniques to derive effectiveacoustic equations for asymptotically small ε. This process turns out tointroduce new memory effects. The effective material parameters are determined from thesolution of frequency-dependent micro-structure cell problems. We propose a numericalapproach to investigate the sound propagation in the homogenized parenchyma using aDiscontinuous Galerkin formulation. Numerical examples are presented.

The method of choice for describing attractive quantum systems is Hartree−Fock−Bogoliubov(HFB) theory. This is a nonlinear model which allows for the description ofpairing effects, the main explanation for the superconductivity ofcertain materials at very low temperature. This paper is the first study ofHartree−Fock−Bogoliubov theory from the point of view of numerical analysis. We start bydiscussing its proper discretization and then analyze the convergence of the simple fixedpoint (Roothaan) algorithm. Following works by Cancès, Le Bris and Levitt for electrons inatoms and molecules, we show that this algorithm either converges to a solution of theequation, or oscillates between two states, none of them being solution to the HFBequations. We also adapt the Optimal Damping Algorithm of Cancès and Le Bris to the HFBsetting and we analyze it. The last part of the paper is devoted to numerical experiments.We consider a purely gravitational system and numerically discover that pairing alwaysoccurs. We then examine a simplified model for nucleons, with an effective interactionsimilar to what is often used in nuclear physics. In both cases we discuss the importanceof using a damping algorithm.

The aim of this work is to present a computationally efficient algorithm to simulate thedeformations suffered by a viscoplastic body in a solidification process. This type ofproblems involves a nonlinearity due to the considered thermo-elastic-viscoplastic law. Inour previous papers, this difficulty has been solved by means of a duality method, knownas Bermúdez–Moreno algorithm, involving a multiplier which was computed with a fixed pointalgorithm or a Newton method. In this paper, we will improve the former algorithms bymeans of a generalized duality method with variable parameters and we will presentnumerical results showing the applicability of the resultant algorithm to solidificationprocesses. Furthermore, we will describe a numerical procedure to choose a constantparameter for the Bermúdez–Moreno algorithm which gives good results when it is applied tosolidification processes.

Inspired by the growing use of non linear discretization techniques for the lineardiffusion equation in industrial codes, we construct and analyze various explicit nonlinear finite volume schemes for the heat equation in dimension one. These schemes areinspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I348 (2010) 691–695]. They preserve the maximum principle and admita finite volume formulation. We provide a original functional setting for the analysis ofconvergence of such methods. In particular we show that the fourth discrete derivative isbounded in quadratic norm. Finally we construct, analyze and test a new explicit nonlinear maximum preserving scheme with third order convergence: it is optimal on numericaltests.

Galerkin reduced-order models for the semi-discrete wave equation, that preserve thesecond-order structure, are studied. Error bounds for the full state variables are derivedin the continuous setting (when the whole trajectory is known) and in the discrete settingwhen the Newmark average-acceleration scheme is used on the second-order semi-discreteequation. When the approximating subspace is constructed using the proper orthogonaldecomposition, the error estimates are proportional to the sums of the neglected singularvalues. Numerical experiments illustrate the theoretical results.

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phaseflow model, that is able to cope with arbitrarily small values of the statistical phasefractions. The solver relies on a relaxation approximation of the model for which theRiemann problem is exactly solved for subsonic relative speeds. In an original manner, theRiemann solutions to the linearly degenerate relaxation system are allowed to dissipatethe total energy in the vanishing phase regimes, thereby enforcing the robustness andstability of the method in the limits of small phase fractions. The scheme is proved tosatisfy a discrete entropy inequality and to preserve positive values of the statisticalfractions and densities. The numerical simulations show a much higher precision and a morereduced computational cost (for comparable accuracy) than standard numerical schemes usedin the nuclear industry. Finally, two test-cases assess the good behavior of the schemewhen approximating vanishing phase solutions.

The reduced basis method is a model reduction technique yielding substantial savings ofcomputational time when a solution to a parametrized equation has to be computed for manyvalues of the parameter. Certification of the approximation is possible by means of ana posteriori error bound. Under appropriate assumptions, this errorbound is computed with an algorithm of complexity independent of the size of the fullproblem. In practice, the evaluation of the error bound can become very sensitive toround-off errors. We propose herein an explanation of this fact. A first remedy has beenproposed in [F. Casenave, Accurate a posteriori error evaluation in thereduced basis method. C. R. Math. Acad. Sci. Paris 350(2012) 539–542.]. Herein, we improve this remedy by proposing a new approximationof the error bound using the empirical interpolation method (EIM). This method achieveshigher levels of accuracy and requires potentially less precomputations than the usualformula. A version of the EIM stabilized with respect to round-off errors is also derived.The method is illustrated on a simple one-dimensional diffusion problem and athree-dimensional acoustic scattering problem solved by a boundary element method.

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy algorithm. In a further improved approach, this is combined with an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched. A safety check step is added at the end of the algorithm to certify the quality of the sampling. Both these techniques are applicable to high-dimensional problems and we shall demonstrate their performance on a number of numerical examples.

In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.