This paper proposes and analyzes a BEM-FEM scheme to approximatea time-harmonic diffusion problem in the plane with non-constantcoefficients in a bounded area. The model is set as a Helmholtztransmission problem with adsorption and with non-constantcoefficients in a bounded domain. We reformulate the problem as afour-field system. For the temperature and the heat flux we usepiecewise constant functions and lowest order Raviart-Thomaselements associated to a triangulation approximating the boundeddomain. For the boundary unknowns we take spaces of periodicsplines. We show how to transmit information from the approximateboundary to the exact one in an efficient way and provewell-posedness of the Galerkin method. Error estimates areprovided and experimentally corroborated at the end of the work.

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods fornearly and perfectly incompressible linear elasticity. These mixed methods allow thechoice of polynomials of any order k ≥ 1 for the approximation of thedisplacement field, and of order k or k − 1 for thepressure space, and are stable for any positive value of the stabilization parameter. Weprove the optimal convergence of the displacement and stress fields in both cases, witherror estimates that are independent of the value of the Poisson’s ratio. These estimatesdemonstrate that these methods are locking-free. To this end, we prove the correspondinginf-sup condition, which for the equal-order case, requires a construction to establishthe surjectivity of the space of discrete divergences on the pressure space. In theparticular case of near incompressibility and equal-order approximation of thedisplacement and pressure fields, the mixed method is equivalent to a displacement methodproposed earlier by Lew et al. [Appel. Math. Res. express3 (2004) 73–106]. The absence of locking of this displacementmethod then follows directly from that of the mixed method, including the uniform errorestimate for the stress with respect to the Poisson’s ratio. We showcase the performanceof these methods through numerical examples, which show that locking may appear ifDirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

We study numerically the semiclassical limit for the nonlinearSchrödinger equation thanks to a modification of the Madelungtransform due to Grenier. This approach allows for the presence ofvacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in thesemiclassical limit, with a numerical rate of convergence inaccordance with the theoreticalresults, before shocks appear in the limiting Eulerequation. By using simple projections, the mass and the momentum ofthe solution are well preserved by the numerical scheme,while the variation of the energy is not negligiblenumerically. Experiments suggest that beyond the critical time for theEuler equation, Grenier's approach yields smooth but highlyoscillatory terms.

We propose a multiscale model reduction method for partial differential equations. Themain purpose of this method is to derive an effective equation for multiscale problemswithout scale separation. An essential ingredient of our method is to decompose theharmonic coordinates into a smooth part and a highly oscillatory part so that the smoothpart is invertible and the highly oscillatory part is small. Such a decomposition plays akey role in our construction of the effective equation. We show that the solution to theeffective equation is in H2, and can be approximated by a regularcoarse mesh. When the multiscale problem has scale separation and a periodic structure,our method recovers the traditional homogenized equation. Furthermore, we provide erroranalysis for our method and show that the solution to the effective equation is close tothe original multiscale solution in the H1 norm. Numerical results are presentedto demonstrate the accuracy and robustness of the proposed method for several multiscaleproblems without scale separation, including a problem with a high contrastcoefficient.

In this work, we consider dynamic frictionless contact with adhesionbetween a viscoelastic body of the Kelvin-Voigt type and astationary rigid obstacle, based on the Signorini's contact conditions.Including the adhesion processes modeled by the bonding field, a newversion of energy function is defined. We use the energy functionto derive a new form of energy balance which is supported by numericalresults. Employing the time-discretization, we establish a numerical formulation and investigate the convergence of numerical trajectories. The fullydiscrete approximation which satisfies the complementarity conditionsis computed by using the nonsmooth Newton's method with the Kanzow-Kleinmichelfunction. Numerical simulations of a viscoelastic beam clamped attwo ends are presented.

For a stationary Markov process the detailed balance condition is equivalent to thetime-reversibility of the process. For stochastic differential equations (SDE’s), the timediscretization of numerical schemes usually destroys the time-reversibility property.Despite an extensive literature on the numerical analysis for SDE’s, their stabilityproperties, strong and/or weak error estimates, large deviations and infinite-timeestimates, no quantitative results are known on the lack of reversibility of discrete-timeapproximation processes. In this paper we provide such quantitative estimates by using theconcept of entropy production rate, inspired by ideas from non-equilibrium statisticalmechanics. The entropy production rate for a stochastic process is defined as the relativeentropy (per unit time) of the path measure of the process with respect to the pathmeasure of the time-reversed process. By construction the entropy production rate isnonnegative and it vanishes if and only if the process is reversible. Crucially, from anumerical point of view, the entropy production rate is an a posterioriquantity, hence it can be computed in the course of a simulation as the ergodicaverage of a certain functional of the process (the so-called Gallavotti−Cohen (GC) action functional). We computethe entropy production for various numerical schemes such as explicit Euler−Maruyama and explicit Milstein’s forreversible SDEs with additive or multiplicative noise. In addition we analyze the entropyproduction for the BBK integrator for the Langevin equation. The order (in thetime-discretization step Δt) of the entropy production rate provides a tool toclassify numerical schemes in terms of their (discretization-induced) irreversibility. Ourresults show that the type of the noise critically affects the behavior of the entropyproduction rate. As a striking example of our results we show that the Euler scheme formultiplicative noise is not an adequate scheme from a reversibilitypoint of view since its entropy production rate does not decrease withΔt.


We show that the entropy method, that has been used successfully in orderto prove exponential convergence towards equilibrium with explicit constants in many contexts,among which reaction-diffusion systems coming out of reversible chemistry, can also be usedwhen one considers a reaction-diffusion system corresponding to an irreversible mechanism ofdissociation/recombination, for which no natural entropy is available.


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.

When applied to the linear advection problem in dimension two, the upwind finite volume method is a non consistent scheme in the finite differences sense but a convergent scheme. According to our previous paper [Bouche et al.,SIAM J. Numer. Anal.43 (2005) 578–603], a sufficient condition in order to complete the mathematical analysis of the finite volume scheme consists in obtaining an estimation of order p, less or equal to one, of a quantity that depends only on the mesh and on the advection velocity and that we called geometric corrector. In [Bouche et al., Hermes Science publishing,London, UK (2005) 225–236], we prove that, on the mesh given by Peterson [SIAM J. Numer. Anal.28 (1991) 133–140] and for a subtle alignment of the direction of transport parallel to the vertical boundary, the infinite norm of the geometric corrector only behaves like h 1/2 where h is a characteristic size of the mesh. This paper focuses on the case of an oblique incidence i.e. a transport direction that is not parallel to the boundary, still with the Peterson mesh. Using various mathematical technics, we explicitly compute an upper bound of the geometric corrector and we provide a probabilistic interpretation in terms of Markov processes. This bound is proved to behave like h, so that the order of convergence is one. Then the reduction of the order of convergence occurs only if the direction of advection is aligned with the boundary.

First–order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a (2K+1)-point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.

The aim of this work is to establish, from amathematical point of view, the limit α → +∞ in the system $i \partial_t E+\nabla (\nabla . E)-\alpha^2 \nabla \times\nabla \times E =-|E|^{2\sigma}E,$ where $E:{\ensuremath{{\Bbb R}}}^3\rightarrow{\mathbb C}^3$ . This corresponds to an approximationwhich is made in the context of Langmuir turbulence in plasmaPhysics. The L 2-subcritical σ (that is σ ≤ 2/3)and the H 1-subcritical σ (that is σ ≤ 2) arestudied. In the physical case σ = 1, the limit is then studied for the $H^1({\ensuremath{{\Bbb R}}}^3)$ norm.

In this paper we introduce a coupled systems of kinetic equations forthe linearized Carleman model. We then study the existencetheory andthe asymptotic behaviour of the resulting coupled problem. In order tosolve the coupled problem we propose to use the time marching algorithm.We then develop a convergence theory for the resulting algorithm. Numericalresults confirming the theory are then presented.

We discuss a numerical formulation for the cell problem related to a homogenization approach for the study of wetting on micro rough surfaces. Regularity properties of the solution are described in details and it is shown that the problem is a convex one. Stability of the solution with respect to small changes of the cell bottom surface allows for an estimate of the numerical error, at least in two dimensions. Several benchmark experiments are presented and the reliability of the numerical solution is assessed, whenever possible, by comparison with analytical one. Realistic three dimensional simulations confirm several interesting features of the solution, improving the classical models of study of wetting on roughness.

In this article, we present a numerical scheme based on a finite element method in orderto solve a time-dependent convection-diffusion equation problem and satisfy someconservation properties. In particular, our scheme is able to conserve the total energyfor a heat equation or the total mass of a solute in a fluid for a concentration equation,even if the approximation of the velocity field is not completely divergence-free. Weestablish a priori errror estimates for this scheme and we give some numerical exampleswhich show the efficiency of the method.

In this work, the least pointwise upper and/or lower bounds on the state variableon a specified subdomain of a control system under piecewise constant control action are sought.This results in a non-smooth optimization problem in function spaces. Introducing a Moreau-Yosidaregularization of the state constraints, the problem can be solvedusing a superlinearly convergent semi-smooth Newton method.Optimality conditions are derived, convergence of the Moreau-Yosidaregularization is proved, and well-posedness and superlinearconvergence of the Newton method is shown. Numerical examplesillustrate the features of this problem and the proposed approach.

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping(given in terms of a boundary integral operator) to solve linear exterior transmission problems inthe plane. As a model we consider a second order elliptic equation in divergence form coupled withthe Laplace equation in the exterior unbounded region. We show that the resulting mixed variationalformulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derivethe usual Cea error estimate and the corresponding rate of convergence. In addition, we develop twodifferent a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser typereliable estimates, respectively. Several numerical results illustrate the suitability of theseestimators for the adaptive computation of the discrete solutions.

In this paper, we study a postprocessing procedure for improvingaccuracy of the finite volume element approximations of semilinearparabolic problems. The procedure amounts to solve a source problemon a coarser grid and then solve a linear elliptic problem on afiner grid after the time evolution is finished. We derive errorestimates in the L 2 and H 1 norms for the standard finitevolume element scheme and an improved error estimate in the H 1 norm. Numerical results demonstrate the accuracy and efficiency ofthe procedure.

This work presents an effective and accurate method for determining, from a theoretical and computational point of view, the time-harmonic Green's function of an isotropic elastic half-plane where an impedance boundary condition is considered. This method, based on the previous work done by Durán et al. (cf. [Numer. Math.107 (2007) 295–314; IMA J. Appl. Math.71 (2006) 853–876]) for the Helmholtz equation in a half-plane, combines appropriately analytical and numerical techniques, which has an important advantage because the obtention of explicit expressions for the surface waves. We show, in addition to the usual Rayleigh wave, another surface wave appearing in some special cases. Numerical results are given to illustrate that. This is an extended and detailed version of the previous article by Durán et al. [C. R. Acad. Sci. Paris, Ser. IIB334 (2006) 725–731].

The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version of the Picard iteration method. The accuracy of the proposed algorithm is investigated.


In this paper, a dynamic viscoelastic problem is numerically studied. The variationalproblem is written in terms of the velocity field and it leads to a parabolic linearvariational equation. A fully discrete scheme is introduced by using thefinite element method to approximate the spatial variable andan Euler scheme to discretize time derivatives. An a priori error estimatesresult is recalled, from which the linear convergence is derived under suitableregularity conditions. Then, an a posteriorierror analysis is provided, extending some preliminary resultsobtained in the study of the heat equation and quasistatic viscoelastic problems.Upper and lower error bounds are obtained. Finally, some two-dimensionalnumerical simulations are presented to show the behavior of the error estimators.