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.

In this paper, we investigate the coupling between operator splitting techniques and a timeparallelization scheme, the parareal algorithm,as a numericalstrategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts,spatially very localized.In a series of previous studies, the numerical analysis of theoperator splitting as well as the parareal algorithmhas been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterizationof such techniques for these stiff configurations.Therefore,we conduct in this work a precise numerical analysis that considers thecombination of time operator splitting and the parareal algorithmin the context of stiff reaction fronts. The impact of the stiffness featured by these frontson the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution ofsuch problems.We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimatesand examine the performance of such strategiesin the context of academical one-dimensional test casesas well as multi-dimensional configurationssimulated on parallel architecture.

Several realistic situations in vehicular traffic that give rise to queues can be modeled through conservation laws with boundary and unilateral constraints on the flux. This paper provides a rigorous analytical framework for these descriptions, comprising stability with respect to the initial data, to the boundary inflow and to the constraint. We present a framework to rigorously state optimal management problems and prove the existence of the corresponding optimal controls. Specific cases are dealt with in detail through ad hoc numerical integrations. These are here obtained implementing the wave front tracking algorithm, which appears to be very precise in computing, for instance, the exit times.

We study an atomistic pair potential-energy E (n)(y) that describesthe elastic behavior of two-dimensional crystals with n atoms where $y \in {\mathbb R}^{2\times n}$ characterizes the particle positions. The mainfocus is the asymptotic analysis of the ground state energy as ntends to infinity. We show in a suitable scaling regime where theenergy is essentially quadratic that the energy minimum of E (n)admits an asymptotic expansion involving fractional powers of n:

${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$

The bulk energy density E bulk is given by an explicitexpression involving the interaction potentials. The surface energyE surface can be expressed as a surface integral where theintegrand depends only on the surface normal and the interactionpotentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggestthat the integrand is a continuous, but nowhere differentiable function ofthe surface normal.

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.

We derive an optimal lower bound of theinterpolation error for linear finite elements on a bounded two-dimensionaldomain. Using the supercloseness between the linear interpolantof the true solution of an elliptic problem and its finite elementsolution on uniform partitions, we furtherobtain two-sided a priori bounds of the discretization error by means of theinterpolation error. Two-sided bounds for bilinear finite elementsare given as well. Numerical tests illustrate our theoreticalanalysis.


In this paper, a dynamic viscoelastic problem is numerically studied. The variationalproblem is written in terms of the velocity field and it leads to a parabolic linearvariational equation. A fully discrete scheme is introduced by using thefinite element method to approximate the spatial variable andan Euler scheme to discretize time derivatives. An a priori error estimatesresult is recalled, from which the linear convergence is derived under suitableregularity conditions. Then, an a posteriorierror analysis is provided, extending some preliminary resultsobtained in the study of the heat equation and quasistatic viscoelastic problems.Upper and lower error bounds are obtained. Finally, some two-dimensionalnumerical simulations are presented to show the behavior of the error estimators.

We study the linearized water-wave problem in a bounded domain (e.g. afinite pond of water) of ${\mathbb R}^3$, having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered thatin this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point ${\mathcal O}$ of the water surface, wherea submerged body touches the surface (see Fig. 1). The radiation conditions emerge from the requirement thatthe linear operator associated to the problem be Fredholm of index zeroin relevant weighted function spaces with separated asymptotics.The classification of incoming and outgoing (seen from ${\mathcal O}$) waves and the unitary scattering matrix are introduced.

We study numerically the semiclassical limit for the nonlinearSchrödinger equation thanks to a modification of the Madelungtransform due to Grenier. This approach allows for the presence ofvacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in thesemiclassical limit, with a numerical rate of convergence inaccordance with the theoreticalresults, before shocks appear in the limiting Eulerequation. By using simple projections, the mass and the momentum ofthe solution are well preserved by the numerical scheme,while the variation of the energy is not negligiblenumerically. Experiments suggest that beyond the critical time for theEuler equation, Grenier's approach yields smooth but highlyoscillatory terms.