We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples.

In this paper, we study how solutions to elliptic problems withperiodically oscillating coefficients behave inthe neighborhood of the boundary of a domain. We extend theresults known for flat boundaries to domains with curved boundariesin the case of a layered medium. This is done by generalizing thenotion of boundary layer and by defining boundary correctors whichlead to an approximation of order ε in the energy norm.

In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uE of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uE based on a multiscale superposition of the unperturbed solution u 0 and a profile defined in a model domain. We conclude with numerical results.

A continuous finite element method to approximate Friedrichs' systems isproposed and analyzed. Stability is achieved by penalizing the jumpsacross meshinterfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence ratesin the graph norm and suboptimal of order ½ convergence rates inthe L 2-norm. A variant of the method specialized toFriedrichs' systems associated with elliptic PDE's in mixed form andreducing the number of nonzero entries in the stiffness matrix is alsoproposed and analyzed. Finally, numerical results are presented to illustrate thetheoretical analysis.

This paper is devoted to the study of a turbulentcirculation model. Equations are derived from the “Navier-Stokes turbulentkinetic energy” system. Some simplifications are performed but attentionis focused on non linearities linked to turbulent eddy viscosity  $\nu _{t}$ . The mixing length $\ell $ acts as a parameter which controls theturbulent part in $\nu _{t}$ . The main theoretical results that we haveobtained concern the uniqueness of the solution for bounded eddy viscositiesand small values of $\ell $ and its asymptotic decreasing as $\ell\rightarrow \infty $ in more general cases. Numerical experimentsillustrate but also allow to extend these theoretical results: uniqueness isproved only for $\ell $ small enough while regular solutions are numericallyobtained for any values of $\ell $ . A convergence theorem is proved forturbulent kinetic energy: $k_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty ,$ but for velocity $u_{\ell }$ we obtain only weaker results.Numerical results allow to conjecture that $k_{\ell }\rightarrow 0,$ $\nu_{t}\rightarrow \infty $ and $u_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty .$ So we can conjecture that this classical turbulent model obtainedwith one degree of closure regularizes the solution.

In a recent paper [4] we have proposed and analyseda suitable mathematical modelwhich describes the coupling of the Navier-Stokes with theOseen equations.In this paper we propose a numerical solution of the coupledproblem by subdomain splitting.After a preliminary analysis, we prove a convergence result foran iterative algorithm that alternates the solution of the Navier-Stokes problem to the one of the Oseen problem.

We develop a well-posedness theory for second order systems in bounded domains whereboundary phenomena like glancing and surface waves play an important role. Attempts havepreviously been made to write a second order system consisting of nequations as a larger first order system. Unfortunately, the resulting first order systemconsists, in general, of more than 2n equations which leads to manycomplications, such as side conditions which must be satisfied by the solution of thelarger first order system. Here we will use the theory of pseudo-differential operatorscombined with mode analysis. There are many desirable properties of this approach: (1) thereduction to first order systems of pseudo-differential equations poses no difficulty andalways gives a system of 2n equations. (2) We can localize the problem,i.e., it is only necessary to study the Cauchy problem and halfplaneproblems with constant coefficients. (3) The class of problems we can treat is much largerthan previous approaches based on “integration by parts”. (4) The relation betweenboundary conditions and boundary phenomena becomes transparent.

The fluctuation splitting schemes were introduced by Roe in the beginning of the80's and have been then developed since then, essentially thanks to Deconinck.In this paper, the fluctuation splittingschemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of thethree dimensional Euler equations is presented. This decomposition leads to an acousticsubsystem and two scalar advection equations, one of them being the entropy advection.Thanks to this decomposition, the two scalar equations are treated with the well knownPSI scalar fluctuation splitting scheme, and the acoustic subsystem is treatedwith the Lax Wendroff matrix fluctuation splitting scheme. Anadditional viscous term is introduced in order to reduce the oscillatory behavior of theLax Wendroff scheme. An implicit form leadsto a robust scheme which enables computations over a large rangeof Mach number. This fluctuation splitting scheme, called the Lax Wendroff - PSI scheme, produceslittlespurious entropy, thus leading to accurate drag predictions.Numerical results obtained with this Lax Wendroff PSI scheme are presented and compared to areference Euler code, based on a Lax Wendroff scheme.

We consider a dynamical one-dimensional nonlinear von Kármán model for beamsdepending on a parameter ε > 0 and studyits asymptotic behavior for t large, as ε → 0. Introducing appropriate dampingmechanisms we show that the energy of solutionsof the corresponding damped models decayexponentially uniformly with respect to theparameter ε. In order for this to be true thedamping mechanism has to have the appropriatescale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko beam modelsfor which the energy tends to zero exponentiallyas well. This is done both in the case ofinternal and boundary damping. We address the sameproblem for plates with internal damping.

In this paper, we present numerical methodsfor the determination of solitons, that consist in spatially localizedstationary states of nonlinear scalar equations or coupled systemsarising in nonlinear optics.We first use the well-known shooting method in order to findexcited states (characterized by the number k of nodes) for theclassical nonlinear Schrödinger equation. Asymptotics can thenbe derived in the limits of either large k are large nonlinear exponents σ.In a second part, we compute solitons for a nonlinearsystem governing the propagation of two coupled waves in a quadratic media in anyspatial dimension, starting from one-dimensional states obtained with a shooting method and considering the dimension as a continuation parameter. Finally, we investigate the case of three wavemixing, for which the shooting method is not relevant.

We consider general surface energies, which areweighted integrals over a closed surface with a weight functiondepending on the position, the unit normal andthe mean curvature of the surface. Energiesof this form have applications in many areas, such as materials science,biology and image processing. Often one is interested in findinga surface that minimizes such an energy, which entails finding its firstvariation with respect to perturbations of the surface.We present a concise derivation of the first variation of thegeneral surface energy using tools from shape differential calculus.We first derive a scalar strong form and nexta vector weak form of the first variation. The latter reveals thevariational structure of the first variation, avoids dealingexplicitly with the tangential gradient of the unit normal,and thus can be easily discretized using parametric finite elements.Our results are valid for surfaces in any number of dimensionsand unify all previous results derived for specific examples ofsuch surface energies.

In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order reduction

In this work, we consider the quasistatic frictionless contact problem between aviscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelasticconstitutive law is employed to model the piezoelectric material and the normal compliancecondition is used to model the contact. The variational formulation is derived in a formof a coupled system for the displacement and electric potential fields. An existence anduniqueness result is recalled. Then, a fully discrete scheme is introduced based on thefinite element method to approximate the spatial variable and an Euler scheme to discretizethe time derivatives. Error estimates are derived on the approximative solutions and,as a consequence, the linear convergence of the algorithm is deduced under suitableregularity conditions. Finally, some two-dimensional examples are presented to demonstratethe performance of the algorithm.

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to the small parameter ε and obtain optimal bound for the modelling error outside the sheet of order $\varepsilon^{N+1}$ for the condition of order N. We end the paper with numerical experiments involving high order finite elements for sheets with varying curvature.

This paper focuses on a one-dimensional wave equation being subjected to a unilateralboundary condition. Under appropriate regularity assumptions on the initial data, a newproof of existence and uniqueness results is proposed. The mass redistribution method,which is based on a redistribution of the body mass such that there is no inertia at thecontact node, is introduced and its convergence is proved. Finally, some numericalexperiments are reported.

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 present the semi-conductor Boltzmann equation,which is time-reversible, and indicate that it can be formally derivedby considering the large time and small perturbing potential limitin the Von-Neumann equation (time-reversible). We then rigorously computethe corresponding asymptotics in the case of the Von-Neumann equation onthe Torus. We show that the limiting equation we obtain does not coincidewith the physically realistic model. The former is indeedan equation of Boltzmann type, yet with memory in time, so that it appearsto be time-reversible. We comment on this point, and further describethe properties of the limiting equation.

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density ϕ. Their time-evolution leads to a nonlinear wave equation $u_{tt}=\mbox{div}\:S(Du)$ with the non-monotone stress-strain relation $S=D\phi$ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very weak sense. It is shown that discrete solutions exist and generate weakly convergent subsequences whose limit is a Young-measure solution. Numerical examples in one space dimension illustrate the time-evolving phase transitions and microstructures of a nonlinearly vibrating string.

We study a free energy computation procedure, introduced in[Darve and Pohorille, J. Chem. Phys.115 (2001) 9169–9183; Hénin and Chipot, J. Chem. Phys.121 (2004) 2904–2914], which relies on the long-timebehavior of a nonlinear stochasticdifferential equation. This nonlinearity comes from a conditionalexpectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions tothis equation has been provedin [Lelièvre et al., Nonlinearity21 (2008) 1155–1181], under some existence and regularity assumptions.In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear equation, andwe study a particle approximation technique based on a Nadaraya-Watson estimator ofthe conditional expectation. The particle system converges to the solutionof the nonlinear equation if the number of particles goes to infinityand then the kernel used in the Nadaraya-Watson approximation tends to aDirac mass. We derive a rate for this convergence, and illustrate it by numericalexamples on a toy model.