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.

The aim of this work is to present a computationally efficient algorithm to simulate thedeformations suffered by a viscoplastic body in a solidification process. This type ofproblems involves a nonlinearity due to the considered thermo-elastic-viscoplastic law. Inour previous papers, this difficulty has been solved by means of a duality method, knownas Bermúdez–Moreno algorithm, involving a multiplier which was computed with a fixed pointalgorithm or a Newton method. In this paper, we will improve the former algorithms bymeans of a generalized duality method with variable parameters and we will presentnumerical results showing the applicability of the resultant algorithm to solidificationprocesses. Furthermore, we will describe a numerical procedure to choose a constantparameter for the Bermúdez–Moreno algorithm which gives good results when it is applied tosolidification processes.

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.

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.

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.

Lower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for tilted perforations. The main techniques are a proper use of the Dirichlet and Kelvin variational principlesin the context of Beppo-Levi spaces. The derivations are validated by numerical experiments in 2D for the axisymmetric case as well as for the full three-dimensional problem.

Inequalities for spatial competition verify the pair approximation of statistical mechanics introduced to theoretical ecology by Matsuda, Satō and Iwasa, among others. Spatially continuous moment equations were introduced by Bolker and Pacala and use a similar assumption in derivation. In the present article, I prove upper bounds for the $k\mathrm{th} $ central moment of occupied sites in the contact process of a single spatial dimension. This result shows why such moment closures are effective in spatial ecology.

This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of a multiobjective dynamic programming equation. The epigraph of this approximate set-valued return function equals to the finite discrete viability kernel resulting from the convergent numerical approximation of the viability kernel proposed in [P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre. Birkhauser, Boston (1999) 177–247. P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Set-Valued Analysis 8 (2000) 111–126]. As a result, the epigraph of the approximate set-valued return function converges to the epigraph of V. The approximate set-valued return function finally provides the proposed numerical approximation of the Pareto optimal set for every initial time and state. Several numerical examples illustrate our approach.

We propose and analyse a method based on the Riccati transformation for solving the evolutionary Hamilton–Jacobi–Bellman equation arising from the dynamic stochastic optimal allocation problem. We show how the fully nonlinear Hamilton–Jacobi–Bellman equation can be transformed into a quasilinear parabolic equation whose diffusion function is obtained as the value function of a certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence and uniqueness and derive useful bounds of classical Hölder smooth solutions. Furthermore, we construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit travelling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 index as an example of the application of the method.

This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. To illustrate some possible applications, an example of linear-quadratic nonzero-sum differential games is worked out. Applying stochastic filtering techniques, we obtain an explicit expression of the equilibrium point.

We study the sequence of monic polynomials orthogonal with respect to inner product

$$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$

$\alpha \gt - 1$

$M\geq 0$

$N\geq 0$

$\zeta \lt 0$

$p$

$q$

A new class of nonparametric nonconforming quadrilateral finite elements is introducedwhich has the midpoint continuity and the mean value continuity at the interfaces ofelements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D.Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] Theparametric DSSY element for general quadrilaterals requires five degrees of freedom tohave an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen andX. Ye, Calcolo 37 (2000) 253–254.], while the newnonparametric DSSY elements require only four degrees of freedom. The design of newelements is based on the decomposition of a bilinear transform into a simple bilinear mapfollowed by a suitable affine map. Numerical results are presented to compare the newelements with the parametric DSSY element.

In this paper we consider a model shape optimization problem. The state variable solvesan elliptic equation on a domain with one part of the boundary described as the graph of acontrol function. We prove higher regularity of the control and develop a priorierror analysis for the finite element discretization of the shape optimizationproblem under consideration. The derived a priori error estimates areillustrated on two numerical examples.

We consider the development and analysis of local discontinuous Galerkin methods forfractional diffusion problems in one space dimension, characterized by having fractionalderivatives, parameterized by β ∈[1, 2]. After demonstrating that aclassic approach fails to deliver optimal order of convergence, we introduce a modifiedlocal numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pureadvection (β = 1) and pure diffusion (β = 2). In the twoclassic limits, known schemes are recovered. We discuss stability and present an erroranalysis for the space semi-discretized scheme, which is supported through a fewexamples.

Taking the cue from stabilized Galerkin methods for scalar advection problems, we adaptthe technique to boundary value problems modeling the advection of magnetic fields. Weprovide rigorous a priori error estimates for both fully discontinuouspiecewise polynomial trial functions and -conforming finite elements.

We consider the flow of a viscous incompressible fluid in a rigid homogeneous porousmedium provided with mixed boundary conditions. Since the boundary pressure can presenthigh variations, the permeability of the medium also depends on the pressure, so that themodel is nonlinear. A posteriori estimates allow us to omit thisdependence where the pressure does not vary too much. We perform the numerical analysis ofa spectral element discretization of the simplified model. Finally we propose a strategywhich leads to an automatic identification of the part of the domain where the simplifiedmodel can be used without increasing significantly the error.

We address the issue of parameter variations in POD approximations of time-dependentproblems, without any specific restriction on the form of parameter dependence.Considering a parabolic model problem, we propose a POD construction strategy allowing usto obtain some a priori error estimates controlled by the POD remainder –in the construction procedure – and some parameter-wise interpolation errors for the modelsolutions. We provide a thorough numerical assessment of this strategy with theFitzHugh − Nagumo 1D model. Finally, we give detailed illustrations of the approach in twoparameter estimation applications, the first in a variational estimation framework withthe FitzHugh − Nagumo model, and the second with a beating heart mechanical model forwhich we employ a sequential estimation method to characterize model parameters using realimage data in a clinical case.

We consider a class of eigenvalue problems for polyharmonic operators, includingDirichlet and buckling-type eigenvalue problems. We prove an analyticity result for thedependence of the symmetric functions of the eigenvalues upon domain perturbations andcompute Hadamard-type formulas for the Frechét differentials. We also considerisovolumetric domain perturbations and characterize the corresponding critical domains forthe symmetric functions of the eigenvalues. Finally, we prove that balls are criticaldomains.

We study the value of European security derivatives in the Black–Scholes model when the underlying asset $\xi $ is approximated by random walks ${\xi }^{(n)} $. We obtain an explicit error formula, up to a term of order $ \mathcal{O} ({n}^{- 3/ 2} )$, which is valid for general approximating schemes and general payoff functions. We show how this error formula can be used to find random walks ${\xi }^{(n)} $ for which option values converge at a speed of $ \mathcal{O} ({n}^{- 3/ 2} )$.