In this paper, we consider C 1,1 Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the SDH method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [Appl. Numer. Math. 45 (2003) 411–418], and we prove it is convergent, and that it preserves the energy and the volume.

We describe a constructive algorithm for obtaining smoothsolutions of a nonlinear, nonhyperbolic pair of balance lawsmodeling incompressible two-phase flow in one space dimension andtime. Solutions are found as stationary solutions of a relatedhyperbolic system, based on the introduction of an artificial timevariable.As may be expected for such nonhyperbolic systems, in general thesolutions obtained do not satisfy both components of the giveninitial data. This deficiency may be overcome, however, byintroducing an alternative “solution" satisfying both componentsof the initial data and an approximate form of a correspondinglinearized system.

We obtain, for entire functions of exponential type satisfying certainintegrability conditions, a quadrature formula using the zeros of sphericalBessel functions as nodes. We deduce from this quadrature formula aresult of Olivier and Rahman, which refines itself a formula of Boas.

We study a microfluidic flow model where the movement of several charged species is coupled with electric field and the motion of ambient fluid. The main numerical difficulty in this model is the net charge neutrality assumption which makes the system essentially overdetermined. Hence we propose to use the involutive and the associated augmented form of the system in numerical computations. Numerical experiments on electrophoresis and stacking show that the completed system significantly improves electroneutrality constraint conservation and recovers analytical results while a direct implementation of the initial model fails.

In this paper, we discuss an hp-discontinuous Galerkin finiteelement method (hp-DGFEM) for the laser surface hardening ofsteel, which is a constrained optimal control problem governed by asystem of differential equations, consisting of an ordinarydifferential equation for austenite formation and a semi-linearparabolic differential equation for temperature evolution. The spacediscretization of the state variable is done using an hp-DGFEM,time and control discretizations are based on a discontinuousGalerkin method. A priori error estimates are developed atdifferent discretization levels. Numerical experimentspresented justify the theoretical order of convergence obtained.

In this paper, a phase field system of Penrose–Fife type withnon–conserved order parameter is considered. A class of time–discrete schemes for an initial–boundary value problem for this phase–field system is presented.In three space dimensions, convergence is proved and an error estimate linear with respect to the time–step sizeis derived.

In this work, depending on the relation between the Deborah, the Reynolds and the aspectratio numbers, we formally derived shallow-water type systems starting from a micro-macrodescription for non-Newtonian fluids in a thin domain governed by an elastic dumbbell typemodel with a slip boundary condition at the bottom. The result has been announced by theauthors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl.Springer Verlag (2010)] and in the present paper, we provide a self-containeddescription, complete formal derivations and various numerical computations. Inparticular, we extend to FENE type systems the derivation of shallow-water models forNewtonian fluids that we can find for instance in [J.-F. Gerbeau, B. Perthame,Discrete Contin. Dyn. Syst. (2001)] which assume an appropriaterelation between the Reynolds number and the aspect ratio with slip boundary condition atthe bottom. Under a radial hypothesis at the leading order, for small Deborah number, wefind an interesting formulation where polymeric effect changes the drag term in the secondorder shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discussintermediate Deborah number with a fixed Reynolds number where a strong coupling is foundthrough a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formallevel, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] includingnon-linear effects (shallow-water framework).

During the development of a parallel solver for Maxwell equations by integral formulations and Fast Multipole Method (FMM), we needed to optimize a critical part including a lot of communications and computations. Generally, many parallel programs need to communicate, but choosing explicitly the way and the instant may decrease the efficiency of the overall program. So, the overlapping of computations and communications may be a way to reduce this drawback. We will see a implementation of this techniques using dynamic and adaptive overlapping based on the EasyMSG high level C++ library over MPI, a case of SPMD programming.

The stability of flat interfaces with respect to a spatial semidiscretization of a solidification model is analyzed. The considered model is the quasi-static approximation of the Stefan problem with dynamical Gibbs–Thomson law. The stability analysis bases on an argument developed by Mullins and Sekerka for the undiscretized case. The obtained stability properties differ from those with respect to the quasi-static model for certain parameter values and relatively coarse meshes. Moreover, consequences on discretization issues are discussed.

The topic of this paper is the numerical analysis of timeperiodic solution for electro-magnetic phenomena. The Limit Absorption Method (LAM)which forms the basis of our study is presented. Theoreticalresults have been proved in the linear finite dimensional case. Thismethod is applied to scattering problems and transport of chargedparticles.

We consider a simple model for the immune systemin which virus are able to undergo mutations and are in competitionwith leukocytes. These mutations are related to several other concepts which havebeen proposed in the literature like those of shape or ofvirulence – a continuous notion. For a given species, the system admits aglobally attractive critical point. We prove that mutations do not affect thispicture for small perturbations and under strong structural assumptions.Based on numerical and theoretical arguments, we also examine how, releasing these assumptions, the system can blow-up.

We present in this paper a stability study concerning finite volume schemesapplied to the two-dimensional Maxwell system, using rectangular or triangular meshes. A stability condition is proved for thefirst-order upwind scheme on a rectangular mesh. Stability comparisonsbetween the Yee scheme and the finite volume formulation are proposed.We also compare the stability domains obtained when considering theMaxwell system and the convection equation.

The present paper proposes and analyzes a general locking free mixed strategy for computing the deformation of incompressible three dimensional structures placed inside flexible membranes. The model involves as in Chapelle and Ferent [Math. Models Methods Appl. Sci.13 (2003) 573–595] a bending dominated shell envelope and a quasi incompressible elastic body. The present work extends an earlier work ofArnold and Brezzi [Math Comp.66 (1997) 1–14] treating the shell part and proposes a global stable finite element approximation by coupling optimal mixed finite element formulations of the different subproblems by mortar techniques. Examples of adequate finite elements are proposed. Convergence results are derived in two steps. First a global inf-sup condition is proved, deducedfrom the local conditions to be satisfied by the finite elements used for the external shell problem, the internal incompressible 3D problem, and the mortar coupling, respectively. Second, the analysis ofArnold and Brezzi [Math. Comp.66 (1997) 1–14] is extended to the present problem and least to convergence results for the full coupled problem, withconstants independent of the problem's small parameters.

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditionsto obtain fast convergence of domain decomposition methods.After explaining the theoretical results, we explicitly compute the coefficients in the transmission boundary conditions. The numerical results presented in this paper confirm the optimality properties.

Evolution equations featuring nonlinearity, dispersion anddissipation are considered here. For classes of such equationsthat include the Korteweg-de Vries-Burgers equation and the BBM-Burgers equation, the zero dissipation limit is studied.Uniform bounds independent of the dissipation coefficient are derived and zero dissipation limit results with optimal convergence rates are established.

In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, δ, to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky [5]. A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html

Many problems in quantumchemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very largedimension of the systems to be solved. Indeed these eigenfunctions dependon 3n variables where n stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed manyinteresting ideas. Among those stands the adiabatic variable method; we present in this paper a mathematical analysis of thisapproximation and propose, in particular, an a posteriori estimate thatmight allow for verifying the adiabaticity hypothesis that is done on some variables; numerical simulations that support thea posteriori estimators obtained theoretically are also presented.

We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with $\gamma=\displaystyle{{c_{p}}/{c_{v}}}=2$ . These equations also modelizethe shallow water problem in height-flow rate formulation used tosolve the flow in lakes and perfectly well-mixed sea. We establisha convergence result for the time-discretized problem when themomentum equation and the continuity equation are solved with theGalerkin method, without adding a penalization term in thecontinuity equation as it is made in Lions (1998). The secondpart is devoted to the numerical analysis and mainly deals withproblems of geophysical fluids. We compare the simulationsobtained with this compressible isentropic Navier–Stokes model andthose obtained with a shallow water model (Di Martino et al., 1999). At first,the computations are executed on a simplified domain in order tovalidate the method by comparison with existing numerical resultsand then on a real domain: the dam of Calacuccia (France). At last, we numerically implement an analyticalexample presented by Weigant (1995) which shows thateven if the data are rather smooth, we cannot have bounds onρ in Lp for p large if $\gamma<2$ when N=2.

This work is concerned with the asymptotic analysis of a time-splitting scheme for theSchrödinger equation with a random potential having weak amplitude, fast oscillations intime and space, and long-range correlations. Such a problem arises for instance in thesimulation of waves propagating in random media in the paraxial approximation. Thehigh-frequency limit of the Schrödinger equation leads to different regimes depending onthe distance of propagation, the oscillation pattern of the initial condition, and thestatistical properties of the random medium. We show that the splitting scheme capturesthese regimes in a statistical sense for a time stepsize independent of the frequency.