A variant of the Total OverlappingSchwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1Math.336(2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains.That same method turns to be an efficient toolto make numerical zoomsin regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture andthe reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper isto use this modified Schwarz procedure as a preconditioner to Krylovsubspaces methods so to accelerate the calculations. A detailed study concludes toa super-linear convergence ofGMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numericalexamples are also provided and commented that demonstrate the reliability of theTOS-preconditioner.

This study is mainly dedicated to the development and analysis ofnon-overlapping domain decomposition methods for solving continuous-pressurefinite element formulations of the Stokes problem. These methods have thefollowing special features. By keeping the equations and unknowns unchanged atthe cross points, that is, points shared by more than two subdomains, one caninterpret them as iterative solvers of the actual discrete problem directlyissued from the finite element scheme. In this way, the good stabilityproperties of continuous-pressure mixed finite element approximations of theStokes system are preserved. Estimates ensuring that each iteration can beperformed in a stable way as well as a proof of the convergence of theiterative process provide a theoretical background for the application of therelated solving procedure. Finally some numerical experiments are given todemonstrate the effectiveness of the approach, and particularly to compare itsefficiency with an adaptation to this framework of a standard FETI-DP method.

In this paper, we discuss an hp-discontinuous Galerkin finiteelement method (hp-DGFEM) for the laser surface hardening ofsteel, which is a constrained optimal control problem governed by asystem of differential equations, consisting of an ordinarydifferential equation for austenite formation and a semi-linearparabolic differential equation for temperature evolution. The spacediscretization of the state variable is done using an hp-DGFEM,time and control discretizations are based on a discontinuousGalerkin method. A priori error estimates are developed atdifferent discretization levels. Numerical experimentspresented justify the theoretical order of convergence obtained.

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.

In this work, the least pointwise upper and/or lower bounds on the state variableon a specified subdomain of a control system under piecewise constant control action are sought.This results in a non-smooth optimization problem in function spaces. Introducing a Moreau-Yosidaregularization of the state constraints, the problem can be solvedusing a superlinearly convergent semi-smooth Newton method.Optimality conditions are derived, convergence of the Moreau-Yosidaregularization is proved, and well-posedness and superlinearconvergence of the Newton method is shown. Numerical examplesillustrate the features of this problem and the proposed approach.

We study the approximation properties of some finite element subspaces ofH(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. Thiswork extends results previously obtained for quadrilateral H(div;Ω) finiteelements and for quadrilateral scalar finite element spaces. The finiteelement spaces we consider are constructed starting from a given finitedimensional space of vector fields on the reference cube, which is thentransformed to a space of vector fields on a hexahedron using the appropriatetransform (e.g., the Piola transform) associated to a trilinear isomorphism ofthe cube onto the hexahedron. After determining what vector fields are neededon the reference element to insure O(h) approximation in L 2(Ω) andin H(div;Ω) and H(curl;Ω) on the physical element, we study the properties ofthe resulting finite element spaces.

In this article, we investigate numerical schemes for solvinga three component Cahn-Hilliard model. The space discretization isperformed by usinga Galerkin formulation and the finite element method.Concerning the time discretization,the main difficulty is to write a scheme ensuring,at the discrete level, the decrease of the free energyand thus the stability of the method.We study three different schemes and proveexistence and convergence theorems. Theoretical results areillustrated by various numerical examples showing that the new semi-implicitdiscretization that we propose seems to be a good compromise between robustnessand accuracy.

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 propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to the small parameter ε and obtain optimal bound for the modelling error outside the sheet of order $\varepsilon^{N+1}$ for the condition of order N. We end the paper with numerical experiments involving high order finite elements for sheets with varying curvature.

We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly improves accuracy in simulations. Standard finite element schemes for NS-α suffer from two major sources of error if their solutions are considered approximations to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error on the velocity error that arises from the (necessary) use of the rotational form nonlinearity. The proposed scheme “fixes” these two numerical issues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We prove the scheme is stable, optimally convergent, and the effect of the pressure error on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate the effectiveness of the method.

The Bidomain model is nowadays one of the most accurate mathematical descriptions of the action potential propagation in the heart. However, its numerical approximation is in general fairly expensive as a consequence of the mathematical features of this system. For this reason, a simplification of this model, called Monodomain problem is quite often adopted in order to reduce computational costs. Reliability of this model is however questionable, in particular in the presence of applied currents and in the regions where the upstroke or the late recovery of the action potential is occurring. In this paper we investigate a domain decomposition approach for this problem, where the entire computational domain is suitably split and the two models are solved in different subdomains. Since the mathematical features of the two problems are rather different, the heterogeneous coupling is non trivial. Here we investigate appropriate interface matching conditions for the coupling on non overlapping domains. Moreover, we pursue an Optimized Schwarz approach for the numerical solution of the heterogeneous problem. Convergence of the iterative method is analyzed by means of a Fourier analysis. We investigate the parameters to be selected in the matching radiation-type conditions to accelerate the convergence. Numerical results both in two and three dimensions illustrate the effectiveness of the coupling strategy.

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

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.

This paper proposes a linear discrete-time scheme for general nonlinear cross-diffusion systems. The scheme can be regarded as an extension of a linear scheme based on the nonlinear Chernoff formula for the degenerate parabolic equations, which proposed by Berger et al. [RAIRO Anal. Numer. 13 (1979) 297–312]. We analyze stability and convergence of the linear scheme. To this end, we apply the theory of reaction-diffusion system approximation. After discretizing the scheme in space, we obtain a versatile, easy to implement and efficient numerical scheme for the cross-diffusion systems. Numerical experiments are carried out to demonstrate the effectiveness of the proposed scheme.

As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed.

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.


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 consider multiscale systems for which only a fine-scalemodel describing the evolution of individuals (atoms,molecules, bacteria, agents) is given, while we are interested in theevolution of the population density on coarse space and timescales. Typically, this evolution is described by a coarseFokker-Planck equation. In this paper, we consider a numerical procedure to compute the solution ofthis Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale,individual-basedsystem. As these parameters might be space- and time-dependent, theestimation is performed in every spatial discretization point and atevery time step. If the fine-scale model is stochastic, the estimationprocedure introduces noise on the coarse level.We investigate stability conditions for this procedure in thepresence of this noise and present ananalysis of the propagation of the estimation error in the numericalsolution of the coarse Fokker-Planck equation.

The standard multilayer Saint-Venant system consists in introducing fluidlayers that are advected by the interfacial velocities. As a consequence there is no massexchanges between these layers and each layer is described by its height and its averagevelocity.Here we introduce another multilayer system with mass exchanges between the neighboringlayers where the unknowns are a total height of water and an average velocity per layer.We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy andhyperbolicity properties of the model. We also give a kinetic interpretation leading toeffective numerical schemes with positivity and energy properties. Numerical tests showthe versatility of the approach and its ability to compute recirculation cases with windforcing.