We show that it is possible to construct a class of entropicschemes for the multicomponent Euler system describing a gas or fluidhomogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. Afirst order Chapman–Enskog expansion shows that the relaxed systemformally converges when the relaxation frequencies go to the infinitytoward a multicomponent Navier–Stokes system with the classical Fick andNewton laws, with a thermal diffusion which can be assimilated to a Soret effect in the case of a fluid mixture,and with also a pressure diffusion or a density diffusion respectively for a gas or fluid mixture. We also discuss on the link between the convexity of the entropies of each species and the existence of the Chapman–Enskog expansion.

We consider a body immersed in a perfect gas and moving under the action of a constant force.Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$|V(t)-V_\infty|\approx \frac{C}{t^{d+1}},$$ where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.

In this paper, we study the long wave approximation for quasilinearsymmetric hyperbolic systems. Using the technics developed byJoly-Métivier-Rauch for nonlinear geometrical optics, we prove thatunder suitable assumptions the long wave limit is described byKdV-type systems. The error estimate if the system is coupled appears tobe better. We apply formally our technics to Euler equations with freesurface and Euler-Poisson systems. This leads to new systems of KdV-type.

We discuss best N-term approximation spaces for one-electron wavefunctions $\phi_i$ and reduced density matrices ρemerging from Hartree-Fock and density functional theory. The approximation spaces $A^\alpha_q(H^1)$ for anisotropicwavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted $\ell_q$ spaces of wavelet coefficients toproof that both $\phi_i$ and ρ are in $A^\alpha_q(H^1)$ for all $\alpha > 0$ with $\alpha = \frac{1}{q} - \frac{1}{2}$ . Our proof is based on the assumption that the $\phi_i$ possess an asymptotic smoothness property at the electron-nuclear cusps.

The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of orderεγ . We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.

We propose and analyze a semi Lagrangian method for the convection-diffusion equation. Error estimates for both semi and fully discrete finite element approximations are obtained for convection dominated flows. The estimates are posed in terms of the projections constructed in [Chrysafinos and Walkington, SIAM J. Numer. Anal. 43 (2006) 2478–2499; Chrysafinos and Walkington, SIAM J. Numer. Anal. 44 (2006) 349–366] and the dependence of various constants upon the diffusion parameter is characterized. Error estimates independent of the diffusion constant are obtained when the velocity field is computed exactly.

This paper is concerned with the analysis and implementation of spectral Galerkinmethods for a class of Fokker-Planck equations that arisesfrom the kinetic theory of dilute polymers. A relevant feature of the class of equationsunder consideration from the viewpoint of mathematical analysis and numerical approximation isthe presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ alongthe boundary ∂D of the computational domain D.Using a symmetrization of the differential operator based on the Maxwellian M corresponding to U,which vanishes along ∂D, we remove the unbounded drift coefficient at the expenseof introducing a degeneracy, through M, in the principal part of the operator.The general class of admissible potentials consideredincludes the FENE (finitely extendible nonlinear elastic) model.We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discretespectral Galerkin method for such degenerate Fokker-Planck equationsthat exhibits optimal-order convergence in the Maxwellian-weighted H1 norm on D.In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with theoriginal operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.

We consider a degenerate parabolic system which modelsthe evolution of nematic liquid crystal with variable degree of orientation.The systemis a slight modificationto that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical fully discrete finite element method for this regularized system, and we establish the (subsequence) convergence of this finite element approximation to the solution of the regularized system as the mesh parameters tend to zero; andto a solution of the original degenerate parabolic system when the the mesh and regularization parameters all approach zero. Finally, numerical experiments are included which show the formation, annihilation and evolution of line singularities/defects in such models.

We consider the effect of surface roughness on solid-solid contact in a Stokes flow.Various models for the roughness are considered, and a unified methodology is given toderive the corresponding asymptotics of the drag force in the close-contact limit. In thisway, we recover and clarify the various expressions that can be found in previousstudies.

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whoseleaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L 1 contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate thecomputational efficiency together with the convergence properties.

The limit behavior of the solutions of Signorini's type-likeproblems in periodically perforated domains with periodε is studied. The main feature of this limit behaviour isthe existence of a critical size of the perforations thatseparates different emerging phenomena as ε → 0. In the critical case, it is shown that Signorini's problemconverges to a problem associated to a new operator whichis the sum of a standard homogenized operator and an extra zeroorder term (“strange term”) coming from the geometry; itsappearance is due to the special size of the holes. The limitproblem captures the two sources of oscillations involved in thiskind of free boundary-value problems, namely, those arising fromthe size of the holes and those due to the periodic inhomogeneityof the medium. The main ingredient of the method used in the proofis an explicit construction of suitable test functions whichprovide a good understanding of the interactions between the abovementioned sources of oscillations.

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of ${\mathbb{R}}^n$. The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion alsotakes place leads to a system of two parabolic equations and one ordinary differential equation coupled with a stiff reaction term. This system is discretized by a finite volume scheme which is suitable for the approximation of thediscontinuous reaction term of unknown sign. Discrete solutions are shown to exist and converge towards a weak solution of the continuous problem. Uniqueness is proved under a Lipschitz condition on the equilibrium gap function. Numerical tests are shown which prove the efficiency of the scheme.

For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.

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.

We introduce and analyse a mixed formulation of theMonge-Kantorovich equations, which express optimality conditions forthe mass transportation problem with cost proportional to distance.Furthermore, we introduce and analyse the finite elementapproximation of this formulation using the lowest orderRaviart-Thomas element. Finally, we present some numericalexperiments, where both the optimal transport density and theassociated Kantorovich potential are computed for a coupling problemand problems involving obstacles and regions of cheaptransportation.

The paper studies the convergence behavior ofMonte Carlo schemes for semiconductors.A detailed analysis of the systematic error with respect to numerical parameters is performed.Different sources of systematic error are pointed out andillustrated in a spatially one-dimensional test case.The error with respect to the number of simulation particlesoccurs during the calculation of the internal electric field.The time step error, which is related to the splitting of transport andelectric field calculations, vanishes sufficiently fast.The error due to the approximation of the trajectories ofparticles depends on the ODE solver used in the algorithm.It is negligible compared to the other sources of time steperror, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanismis the most significant source of error with respect to the time step.

The numerical solution of the Hartree-Fock equations is a central problem in quantumchemistry for which numerous algorithms exist. Attempts to justify these algorithmsmathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod.Numer. Anal. 34 (2000) 749–774], but, to our knowledge, nocomplete convergence proof has been published, except for the large-Zresult of [M. Griesemer and F. Hantsch, Arch. Rational Mech. Anal. (2011)170]. In this paper, we prove the convergence of a natural gradient algorithm, using agradient inequality for analytic functionals due to Łojasiewicz [Ensemblessemi-analytiques. Institut des Hautes Études Scientifiques (1965)]. Then,expanding upon the analysis of [E. Cancès and C. Le Bris, Math. Mod. Numer. Anal.34 (2000) 749–774], we prove convergence results for the Roothaanand Level-Shifting algorithms. In each case, our method of proof provides estimates on theconvergence rate. We compare these with numerical results for the algorithms studied.

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressiblefluid exhibiting phase transitions between a liquid and a vapor phase in the presence ofcapillarity effects close to phase boundaries. Standard numerical discretizations areknown to violate discrete versions of inherent energy inequalities, thus leading tospurious dynamics of computed solutions close to static equilibria (e.g.,parasitic currents). In this work, we propose a time-implicit discretization of theproblem, and use piecewise linear (or bilinear), globally continuous finite element spacesfor both, velocity and density fields, and two regularizing terms where correspondingparameters tend to zero as the mesh-size h > 0 tends to zero.Solvability, non-negativity of computed densities, as well as conservation of mass, and adiscrete energy law to control dynamics are shown. Computational experiments are providedto study interesting regimes of coefficients for viscosity and capillarity.


The Diffusion Monte Carlo method is devoted to the computation ofelectronic ground-state energies of molecules. In this paper, we focus onimplementations of this method which consist in exploring theconfiguration space with a fixed number of random walkers evolvingaccording to a stochastic differential equation discretized in time. Weallow stochastic reconfigurations of the walkers to reduce thediscrepancy between the weights that they carry. On a simpleone-dimensional example, we prove the convergence of the method for afixed number of reconfigurations when the number of walkers tends to+∞ while the timestep tends to 0. We confirm our theoreticalrates of convergence by numerical experiments. Various resamplingalgorithms are investigated, both theoretically and numerically.