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 present two versions of the central localdiscontinuous Galerkin (LDG) method on overlapping cellsfor solving diffusion equations, and provide theirstability analysis and error estimates for the linear heat equation.A comparisonbetween the traditional LDG method ona single mesh and the two versions of the central LDGmethod on overlapping cells is also made.Numerical experiments are provided to validate the quantitativeconclusions from the analysis and to support conclusions forgeneral polynomial degrees.

In this paper, we continue the study of the Raman amplification in plasmas that we initiated in [Colin and Colin, Diff. Int. Eqs.17 (2004) 297–330; Colin and Colin, J. Comput. Appl. Math.193 (2006) 535–562]. We point out that the Raman instability gives rise to three components. The first one is collinear to the incident laser pulse and counter propagates. In 2-D, the two other ones make a non-zero angle with the initial pulse and propagate forward. Furthermore they are symmetric with respect to the direction of propagation of the incident pulse. We construct a non-linear system taking into account all these components and perform some 2-D numerical simulations.

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

The aim of this paper is to develop a finite element method which allows computingthe buckling coefficients and modes of a non-homogeneous Timoshenko beam.Studying the spectral properties of a non-compact operator,we show that the relevant buckling coefficients correspond to isolatedeigenvalues of finite multiplicity.Optimal order error estimates are proved for the eigenfunctionsas well as a double order of convergence forthe eigenvalues using classical abstract spectral approximation theory for non-compact operators.These estimates are valid independently of the thickness of the beam, whichleads to the conclusion that the method is locking-free.Numerical tests are reported in order to assess the performance of the method.

Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ requirethe evaluation of integrals $I = \int_{S^{(1)}}\int_{S^{(2)}}g(x,y){\rm d}y{\rm d}x$ where S (1),S (2) are d-simplices and g has a singularityat x = y. We assume that g is Gevrey smooth for x $\ne$ y andsatisfies bounds for the derivatives which allow algebraic singularitiesat x = y. This holds for kernel functions commonly occurring in integralequations. We construct a family of quadrature rules $\mathcal{Q}_{N}$ usingN function evaluations of g which achieves exponential convergence|I – $\mathcal{Q}_{N}$ | ≤C exp(–r N γ ) with constants r, γ > 0.

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove thatthe scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be appliedto models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as itshigh resolution and robustness in a number of numerical examples.

Cell-centered and vertex-centered finite volume schemes for the Laplace equationwith homogeneous Dirichlet boundary conditionsare considered on a triangular mesh and on the Voronoi diagram associated to its vertices.A broken P 1 function is constructed from the solutions of both schemes.When the domain is two-dimensional polygonal convex,it is shown that this reconstructionconverges with second-order accuracy towards the exact solution in the L 2 norm,under the sufficient condition that the right-hand side of the Laplace equation belongs to H 1(Ω).

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.

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.

We develop a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension. Various properties of the peridynamic operators are examined for general micromodulus functions. These properties are utilized to establish the well-posedness of both the stationary peridynamic model and the Cauchy problem of the time dependent peridynamic model. The connections to the classical elastic models are also provided.

This paper deals with the mathematical and numerical analysis of asimplified two-dimensional model for the interaction between the windand a sail. The wind is modeled as a steady irrotational plane flow pastthe sail, satisfying the Kutta-Joukowski condition. This conditionguarantees that the flow is not singular at the trailing edge of thesail. Although for the present analysis the position of the sail istaken as data, the final aim of this research is to develop tools tocompute the sail shape under the aerodynamic pressure exerted by thewind. This is the reason why we propose a fictitious domain formulationof the problem, involving the wind velocity stream function and aLagrange multiplier; the latter allows computing the force densityexerted by the wind on the sail. The Kutta-Joukowski condition isimposed in integral form as an additional constraint. The resultingproblem is proved to be well posed under mild assumptions. For thenumerical solution, we propose a finite element method based onpiecewise linear continuous elements to approximate the stream functionand piecewise constant ones for the Lagrange multiplier. Error estimatesare proved for both quantities and a couple of numerical testsconfirming the theoretical results are reported. Finally the method isused to determine the sail shape under the action of the wind.

We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional C 2-domains. For solution in Hq for q > 2, we prove that the method constructed with polynomials of degree one on each element approximates the function and its gradient with optimal orders h 2 and h, respectively. When q = 2, we have h 2-ε and h 1-ε for any ϵ > 0 instead. To this end,we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny-Lions' lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to differences between the exact and the approximate domains are treated using Hardy's inequality together with more standard results on Sobolev functions.

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.

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations,we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for the approximation of incompressible flow coupled problems. We exploit stabilized mixed finite elements together with Nitsche's type matching conditions that automatically adapt to the coupling of different combinations of coefficients. We study in details the stability of the method using weighted norms, emphasizing the robustness of the stability estimate with respect to the coefficients. We also consider an iterative method to split the coupled heterogeneous problem in possibly homogeneous local problems, and we investigate the spectral properties of suitable preconditioners for the solution of the global as well as local problems. Finally, we present the simulation of the fully coupled blood flow and plasma filtration problems on a realistic geometry of a cardiovascular artery after the implantation of a drug eluting stent (DES). A similar finite element method for mass transport is then employed to study the evolution of the drug released by the DES in the blood stream and in the arterial walls, and the role of plasma filtration on the drug deposition is investigated.

This article is devoted to the presentation of a new contact algorithm for bodies undergoing finite deformations. We only address the kinematic aspect of the contact problem, that is the numerical treatment of the non-intersection constraint. In consequence, mechanical aspects like friction, adhesion or wear are not investigated and we restrict our analysis to the simplest frictionless case. On the other hand, our method allows us to treat contacts and self-contacts, thin or non-thin structures in a single setting.

We present a convergence analysis of a cell-based finite volume (FV) discretization scheme applied to a problem of control in the coefficients of a generalized Laplace equation modelling, for example, a steady state heat conduction. Such problems arise in applications dealing with geometric optimal design, in particular shape and topology optimization, and are most often solved numerically utilizing a finite element approach. Within the FV framework for control in the coefficients problems the main difficulty we face is the need to analyze the convergence of fluxes defined on the faces of cells, whereas the convergence of the coefficients happens only with respect to the “volumetric” Lebesgue measure. Additionally, depending on whether the stationarity conditions are stated for the discretized or the original continuous problem, two distinct concepts of stationarity at a discrete level arise.We provide characterizations of limit points, with respect to FV mesh size, of globally optimal solutions and two types of stationary points to the discretized problems. We illustrate the practical behaviour of our cell-based FV discretization algorithm on a numerical example.

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutionsof polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ $\mathbb{R}^d$ , d = 2 or 3, for the velocity andthe pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function $\beta^L(\cdot) :=\min(\cdot,L)$ in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker–Planck–Navier–Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions.We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms,and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm.

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticitywith hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effectsand through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent modelis proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful successive limit passage. Computational 3D simulations illustrate an implementation of the method as well as physical effects of residual stresses substantially depending on rate of heat treatment.

In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) $\cap$${\bf \tilde H}$-1/2(div, K)-regularity (r > 0) on the reference element (either triangle or square) K. We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space ${\bf \tilde H}$-1/2(div, K), which is closely related to the energy spaces for boundary integral formulations of time-harmonic problems of electromagnetics in three dimensions.