The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

In this article we develop a posteriori error estimates for second orderlinear elliptic problems with point sources in two- and three-dimensional domains. Weprove a global upper bound and a local lower bound for the error measured in a weightedSobolev space. The weight considered is a (positive) power of the distance to the supportof the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theoryhinges on local approximation properties of either Clément or Scott–Zhang interpolationoperators, without need of modifications, and makes use of weighted estimates forfractional integrals and maximal functions. Numerical experiments with an adaptivealgorithm yield optimal meshes and very good effectivity indices.

We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C.Nguyen and A.T. Patera, An empirical interpolation method: application to efficientreduced-basis discretization of partial differential equations. Compt. Rend. Math.Anal. Num. 339 (2004) 667–672] to a weighted empiricalinterpolation method in order to approximate nonlinear parametric functions with weightedparameters, e.g. random variables obeying various probabilitydistributions. A priori convergence analysis is provided for the proposedmethod and the error bound by Kolmogorov N-width is improved from the recent work [Y.Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolationprocedure: the magic points. Commun. Pure Appl. Anal. 8(2009) 383–404]. We apply our method to geometric Brownian motion, exponentialKarhunen–Loève expansion and reduced basis approximation of non-affine stochastic ellipticequations. We demonstrate its improved accuracy and efficiency over the empiricalinterpolation method, as well as sparse grid stochastic collocation method.

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.

The purpose of this study is the time domain modeling of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependent damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from the horizontal plane. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependent damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings’ inherent difficulties.

We introduce a new stable MINI-element pair for incompressible Stokes equations onquadrilateral meshes, which uses the smallest number of bubbles for the velocity. Thepressure is discretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done withthe standard Q1-conforming elements enriched byone bubble a quadrilateral. The superconvergence in the pressure of the proposed pair isanalyzed on uniform rectangular meshes, and tested numerically on uniform and non-uniformmeshes.

We present a high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.

In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with ε-periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative via a function of order εγ with γ ≤−1. We give the homogenization results which include those obtained by Jose in [Rev. Roum. Math. Pures Appl. 54 (2009) 189–222]. We also get the corrector results.

This work studies the heat equation in a two-phase material with spherical inclusions.Under some appropriate scaling on the size, volume fraction and heat capacity of theinclusions, we derive a coupled system of partial differential equations governing theevolution of the temperature of each phase at a macroscopic level of description. Thecoupling terms describing the exchange of heat between the phases are obtained by usinghomogenization techniques originating from [D. Cioranescu, F. Murat, Collège de FranceSeminar, vol. II. Paris 1979–1980; vol. 60 of Res. Notes Math. Pitman,Boston, London (1982) 98–138].

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.

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusionequation on an interval. The discretization is based on the equation’s gradient flowstructure with respect to the Wasserstein distance. The scheme inherits various propertiesfrom the continuous flow, like entropy monotonicity, mass preservation, metric contractionand minimum/ maximum principles. As the main result, we give a proof of convergence in thelimit of vanishing mesh size under a CFL-type condition. We also present results fromnumerical experiments.

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green’s function by Marahrens and the third author.

Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.

We consider the efficient and reliable solution of linear-quadratic optimal controlproblems governed by parametrized parabolic partial differential equations. To this end,we employ the reduced basis method as a low-dimensional surrogate model to solve theoptimal control problem and develop a posteriori error estimationprocedures that provide rigorous bounds for the error in the optimal control and theassociated cost functional. We show that our approach can be applied to problems involvingcontrol constraints and that, even in the presence of control constraints, the reducedorder optimal control problem and the proposed bounds can be efficiently evaluated in anoffline-online computational procedure. We also propose two greedy sampling procedures toconstruct the reduced basis space. Numerical results are presented to confirm the validityof our approach.

Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases impossible to do within a reasonable time. We propose here a very simple methodology which may accelerate population parametrization of complex models, including partial differential equations models. We illustrate our method on the classical KPP equation.

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C.Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006)717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl.88 (2007) 34–63.]. The equation under consideration is a standardlinear elliptic equation in divergence form, where the highly oscillatory coefficient isthe composition of a periodic matrix with a stochastic diffeomorphism. The homogenizedlimit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions,C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We firstestablish, in the one-dimensional case, a convergence result (with an explicit rate) onthe residual process, defined as the difference between the solution to the highlyoscillatory problem and the solution to the homogenized problem. We next return to themultidimensional situation. As often in random homogenization, the homogenized matrix isdefined from a so-called corrector function, which is the solution to a problem set on theentire space. We describe and prove the almost sure convergence of an approximationstrategy based on truncated versions of the corrector problem.

In this paper we propose and analyze a localized orthogonal decomposition (LOD) methodfor solving semi-linear elliptic problems with heterogeneous and highly variablecoefficient functions. This Galerkin-type method is based on a generalized finite elementbasis that spans a low dimensional multiscale space. The basis is assembled by performinglocalized linear fine-scale computations on small patches that have a diameter of orderH | log (H)| where H is the coarse mesh size. Without any assumptions onthe type of the oscillations in the coefficients, we give a rigorous proof for a linearconvergence of the H1-error with respect to the coarse meshsize even for rough coefficients. To solve the corresponding system of algebraicequations, we propose an algorithm that is based on a damped Newton scheme in themultiscale space.