The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions,depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000,Springer (2010)], the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The present paper complements this work. It is shown that one can reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.

This paper deals with a finite element method to solve fluid-structure interaction problems. More precisely it concerns the numerical computation of harmonic hydroelastic vibrations under gravity. It is based on a displacement formulation for both the fluid and the solid. Gravity effects are included on the free surface of the fluid as wellas on the liquid-solid interface. The pressure of the fluid is used as a variable for the theoretical analysis leading to a well posed mixed linear eigenvalue problem.Lowest order triangular Raviart-Thomas elements are used for the fluid and classical piecewise linear elements for the solid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense yielding a non conforming discretization. The method does not present spurious or circulation modes for nonzero frequencies.Convergence is proved and optimal error estimates are given.Finally, numerical results are shown.

In this article we consider elliptic partial differential equations with random coefficients and/or random forcing terms. In the current treatment of such problems by stochastic Galerkin methods it is standard to assume that the random diffusion coefficient is bounded by positive deterministic constants or modeled as lognormal random field. In contrast, we make the significantly weaker assumption that the non-negative random coefficients can be bounded strictly away from zero and infinity by random variables only and may have distributions different from a lognormal one. We show that in this case the standard stochastic Galerkin approach does not necessarily produce a sequence of approximate solutions that converges in the natural norm to the exact solution even in the case of a lognormal coefficient. By using weighted test function spaces we develop an alternative stochastic Galerkin approach and prove that the associated sequence of approximate solutions converges to the exact solution in the natural norm. Hereby, ideas for the case of lognormal coefficient fields from earlier work of Galvis, Sarkis and Gittelson are used and generalized to the case of positive random coefficient fields with basically arbitrary distributions.

This paper presents anextension to stabilized methods of the standard technique for thenumerical analysis of mixed methods. We prove that the stability of stabilizedmethods follows from an underlying discrete inf-sup condition, plus a uniformseparation property between bubble and velocity finite element spaces. We applythe technique introduced to provethe sta bi li ty of stabilized spectral element methods so asstabilized solution of the primitive equations of the ocean.

In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.

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.

This paper addresses some results on the development of an approximate methodfor computing the acoustic field scattered by a three-dimensional penetrable object immersed into an incompressiblefluid. The basic idea of the method consists in using on-surface differentialoperators that locally reproduce the interior propagation phenomenon. This approach leads tointegral equation formulations with a reduced computational cost compared to standard integral formulations couplingboth the transmitted and scattered waves. Theoreticalaspects of the problem and numerical experiments are reported to analyze the efficiency ofthe method and precise its validity domain.

We present families of scalar nonconforming finite elements of arbitraryorder $r\ge 1$ with optimal approximation properties on quadrilaterals andhexahedra. Their vector-valued versions together with a discontinuouspressure approximation of order $r-1$ form inf-sup stable finite element pairsof order r for the Stokes problem. The well-known elements by Rannacherand Turek are recovered in the case r=1. A numerical comparison betweenconforming and nonconforming discretisations will be given. Since higherorder nonconforming discretisation on quadrilaterals and hexahedra have lessunknowns and much less non-zero matrix entries compared to correspondingconforming methods, these methods are attractive for numerical simulations.

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Q r -elements for the velocity anddiscontinuous $P_{r-1}$ -elements for the pressure where the orderr can vary from element to elementbetween 2 and a fixed bound $r^*$ .We prove the inf-sup condition uniformly with respect to the meshwidth hon general quadrilateral and hexahedral meshes with hanging nodes.

We propose a derivation of a nonequilibrium Langevin dynamics for a large particleimmersed in a background flow field. A single large particle is placed in an ideal gasheat bath composed of point particles that are distributed consistently with thebackground flow field and that interact with the large particle through elasticcollisions. In the limit of small bath atom mass, the large particle dynamics converges inlaw to a stochastic dynamics. This derivation follows the ideas of [P. Calderoni, D. Dürrand S. Kusuoka, J. Stat. Phys. 55 (1989) 649–693. D. Dürr,S. Goldstein and J. Lebowitz, Z. Wahrscheinlichkeit 62(1983) 427–448. D. Dürr, S. Goldstein and J.L. Lebowitz. Comm. Math. Phys.78 (1981) 507–530.] and provides extensions to handle the nonzerobackground flow. The derived nonequilibrium Langevin dynamics is similar to the dynamicsin [M. McPhie, P. Daivis, I. Snook, J. Ennis and D. Evans, Phys. A299 (2001) 412–426]. Some numerical experiments illustrate the useof the obtained dynamic to simulate homogeneous liquid materials under shear flow.

By means of a result on the semi-global C 1 solution, we establish the exact boundary controllability for the reducible quasilinearhyperbolic system if the C 1 norm of initial data and final state issmall enough.

An algorithm for approximation of an unsteady fluid-structure interaction problem isproposed. The fluid is governed by the Navier-Stokes equations with boundary conditionson pressure, while for the structure a particular plate model is used. The algorithm is based on the modal decomposition and the Newmark Method for the structure and on the Arbitrary Lagrangian Eulerian coordinates and the Finite Element Method for the fluid.In this paper, the continuity of the stresses at the interface was treated by theLeast Squares Method. At each time step we have to solve an optimization problemwhich permits us to use moderate time step. This is the main advantage of this approach. In order to solve the optimization problem, we have employed the Broyden, Fletcher, Goldforb, Shano Method wherethe gradient of the cost function was approached by the Finite Difference Method.Numerical results are presented.

In this paper we consider a hyperbolic-parabolic problem that models the longitudinal deformations of a thermoviscoelastic rod supported unilaterally by an elastic obstacle. The existence and uniqueness of a strong solution is shown. A finite element approximation is proposed and its convergence is proved. Numerical experiments are reported.

We consider the classical Interpolating Moving Least Squares (IMLS)interpolant as defined by Lancaster and Šalkauskas [Math. Comp.37 (1981) 141–158] andcompute the first and second derivative of this interpolant at the nodes of agiven grid with the help of a basic lemma on Shepard interpolants. We comparethe difference formulae with those defining optimal finite difference methods anddiscuss their deviation from optimality.

Implicit sampling is a sampling scheme for particle filters, designed to move particles one-by-one so that they remain in high-probability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations.

In this paper we present two-level overlapping domain decomposition preconditioners for the finite-element discretisation of elliptic problems in two and three dimensions. The computational domain is partitioned into overlapping subdomains, and a coarse space correction is added. We present an algebraic way to define the coarse space, based on the concept of aggregation. This employs a (smoothed) aggregation technique and does not require the introduction of a coarse grid. We consider a set of assumptions on the coarse basis functions, to ensure bound for the resulting preconditioned system. These assumptions only involve geometrical quantities associated to the aggregates, namely their diameter and the overlap. A condition number which depends on the product of the relative overlap among the subdomains and the relative overlap among the aggregates is proved. Numerical experiments on a model problem are reported to illustrate the performance of the proposed preconditioners.

For a plate subject to stress boundary condition, the deformationdetermined by the Reissner–Mindlin plate bending model could bebending dominated, transverse shear dominated, or neither(intermediate), depending on the load. We show that theReissner–Mindlin model has a wider range of applicability thanthe Kirchhoff–Love model, but it does not always converge to theelasticity theory. In the case of bending domination, both the twomodels are accurate. In the case of transverse shear domination,the Reissner–Mindlin model is accurate but the Kirchhoff–Lovemodel totally fails. In the intermediate case, while theKirchhoff–Love model fails, the Reissner–Mindlin solution alsohas a relative error comparing to the elasticity solution, whichdoes not decrease when the plate thickness tends to zero. We alsoshow that under the conventional definition of the resultantloading functional, the well known shear correction factor 5/6in the Reissner–Mindlin model should be replaced by 1.Otherwise, the range of applicability of the Reissner–Mindlinmodel is not wider than that of Kirchhoff–Love's.

We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite generalpartial differential operators. The starting point of our analysis is the DPG methodintroduced by [Demkowicz et al., SIAM J. Numer. Anal.49 (2011) 1788–1809; Zitelli et al., J.Comput. Phys. 230 (2011) 2406–2432]. This discretization resultsin a sparse positive definite linear algebraic system which can be obtained from a saddlepoint problem by an element-wise Schur complement reduction applied to the test space.Here, we show that the abstract framework of saddle point problems and domaindecomposition techniques provide stability and a priori estimates. Toobtain efficient numerical algorithms, we use a second Schur complement reduction appliedto the trial space. This restricts the degrees of freedom to the skeleton. We construct apreconditioner for the skeleton problem, and the efficiency of the discretization and thesolution method is demonstrated by numerical examples.

We consider a Boltzmann equation for inelastic particles on the line and prove existence and uniqueness for the solutions.

A Volterra model with mutual interferenceconcerning integrated pest management is proposed and analyzed. Byusing Floquet theorem and small amplitude perturbation method andcomparison theorem, we show the existence of a globallyasymptotically stable pest-eradication periodic solution. Further,we prove that when the stability of pest-eradication periodicsolution is lost, the system is permanent and there exists alocally stable positive periodic solution which arises from thepest-eradication periodic solution by bifurcation theory. When theunique positive periodic solution loses its stability, numericalsimulation shows there is a characteristic sequence ofbifurcations, leading to a chaotic dynamics. Finally, we comparethe validity of integrated pest management (IPM) strategy withclassical methods and conclude IPM strategy is more effective thanclassical methods.