---

SS14 - Analyse numérique / SS14 - Numerical analysis
Org: A. Fortin (Laval) et/and J. Blum (Nice)

---

HABIB AMMARI, Ecole Polytechnique, Palaiseau
Algorithms for anomaly detection

We consider an inverse problem arising in anomaly detection with its mathematical model based on the T-Scan system. We try to detect a small anomaly from measured data that is available only on a small portion of the subject. We carry out rigorous estimates to derive a simple approximation that gives a non-iterative detection algorithm of finding the anomaly. We also present a multi-frequency approach to handle the case where the complex conductivity of the background is not homogeneous and not known a priori. This is a joint work with O. Kwon, J. K. Seo, and E. Woo.

TAHAR ZAMENE BOULMEZAOUD, Université de Pau et des Pays de l'Adour, and Université Paris VI
Inverted Elements: a new method for solving elliptic problems in unbounded domains

In this talk we propose a new numerical method for solving elliptic equations in unbounded regions of space. The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used as a functionnal framework for describing the behavior of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate the performance of the method.

GEORGES-HENRI COTTET, LMC-IMAG, Université Joseph Fourier, Grenoble
Level set methods for fluid-structure interaction problems in 2D and 3D

Level set methods have been designed to capture interfaces with some flexibility in Eulerian formulations of incompressible fluid mechanics. They have been used by Chang et al. with some success for variable density flows with surface tension.

In this talk we propose a level-set method motivated by applications in cell biology which handles elasticity effects on the interface (either curve or surface). This model satisfies energy equalities which is a good sign for consistency. It can also be viewed as an Eulerian alternative of Peskin's immersed boundary methods.

In passing, we also address some technical problems related to interface distortions in the implementation of level-set methods. We propose a simple alternative to the usual reinitialization strategies in general used to overcome these problems.

YVES DEMAY, Nice

ANDRÉ FORTIN, Université Laval, Québec, Canada, G1K 7P4
Remaillage anisotrope et applications au calcul des surfaces libres

Dans cet exposé, nous présenterons une stratégie d'adaptation de maillage pour la résolution de problèmes de mécanique des fluides avec surfaces libres instationnaires. Les maillages obtenus sont caractérisés par une forte anisotropie c.-à-d. les éléments peuvent être fortement allongés dans certaines directions privilégiées de l'écoulement. L'estimation d'erreur est basée sur l'introduction d'une métrique dépendant de la solution.

Les surfaces libres sont calculées par une méthode de surfaces de niveau (level sets) modifiée de manière à assurer une parfaite conservation de la masse. Un algorithme de réinitialisation de la fonction distance associée sera aussi discuté.

Nous présenterons enfin des applications bi et tridimensionnelles à la déformation et éventuellement au bri de gouttelettes de fluides en cisaillement dans un autre fluide.

THIERRY GALLOUET, CMI, 39 rue Joliot-Curie, F-13453 Marseille Cedex 13
Elliptic equations with measure data and numerical schemes

In order to show the existence of solutions for linear elliptic equation, the classical method, due to Stampacchia, is to use a duality argument (and a regularity result for elliptic equations with regular data). Another classical method is to pass to the limit on approximate solutions obtained with regular data (converging towards the measure data). We will present here a third method which consists in passing to the limit on approximate solutions obtained with numerical schemes such as finite element schemes or finite volume schemes. This method is not easier than the previous ones but is interesting since it yields a way to compute approximate solutions. We will also present this method for convection-diffusion equations which lead to noncoercive elliptic equations with measure data.

MARTIN GANDER, McGill University, 805 Sherbrooke Street West, Montreal, H3A 2K6
The Parareal Algorithm in the Context of Classical Methods

A few years ago, Lions, Maday and Turinici introduced a new algorithm for evolution problems which is parallel in time: the parareal algorithm. This algorithm decomposes the time domain into subdomains in time, and then uses fine grid approximations on the time subdomains, and a coarse grid correction in time to iteratively construct a better and better approximation in time of the evolution problem.

I will show in this presentation that the parareal algorithm can be put into the context of more classical methods. In particular, I will show that the parareal algorithm is equivalent to a multigrid waveform relaxation algorithm with a certain choice of smoother, restriction and extension operators, and it is also equivalent to a multiple shooting algorithm with a particular coarse approximation of the Jacobian matrix in the nonlinear Newton solver.

DANIEL LE ROUX, Université Laval
Modes numériques dans les équations de Saint-Venant, l'élément P^NC₁ - P₁

La plupart des méthodes numériques utilisées pour résoudre les équations de Saint-Venant génèrent des modes parasites en approximant les ondes de type inertie-gravité. Les modes parasites les plus dangereux sont les modes stationnaires appartenant aux noyaux des opérateurs gradient, Coriolis et divergence discrets. Dans un premier temps nous passerons ces problèmes en revue en choisissant quelques exemples parmi les grilles de différences finies et d'éléments finis les plus populaires. Dans un deuxième temps, nous présenterons la paire d'éléments finis P^NC₁ -P₁, nous analyserons sa relation de dispersion et nous montrerons que cette paire approxime raisonnablement bien les ondes de type inertie-gravité. Nous la comparerons notamment à la grille C-D en resolvant le probleme de Stommel, c'est à dire le cas d'une circulation forcée par le vent dans un bassin océanique.

YVON MADAY, Labo. J.-L. Lions, Univ. P. et M. Curie
Some variations on the reduced basis method with certification

Reduced basis methods are discretization strategies for the solution of partial differential equations that use "optimal" ad hoc bases constructed in a preliminary stage by standard approximation methods. The number of degrees of freedom required to solve a particular problem is then extremely small compared to the corresponding number involved by conventional methods for similar precision. The range of applications of reduced basis approaches currently goes well beyond the original application of this framework to solid mechanics. All aspects of the reduced-basis approach are not yet mathematically well understood; nevertheless, a posteriori error estimates and validations are now available that provide for rigorous certification of accuracy.

In this talk we shall present some new developments of this method including efficient treatments of nonlinear problems.

BERTRAND MAURY, Laboratoire de Mathématique, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France
Pression granulaire et pression fluide

Nous proposons de donner un aperçu des problèmes posés par l'estimation des multiplicateurs de Lagrange associés à des contraintes unilatérales pour certains problèmes d'évolution du type écoulement granulaire. Nous tâcherons notamment de préciser les liens qui existent entre le calcul des forces de réactions qui empêchent l'interpénétration d'une multitude de corps rigides et le calcul du champ de pression dans la résolution des équations de Stokes ou de Navier-Stokes incompressibles.

NILIMA NIGAM, McGill University, 805 Sherbrooke St. W, Montreal, QC
Perturbative Stekhlov-Poincare maps for general domains

The truncation of an infinite computational domain by an artificial boundary, which arises in the study of exterior scattering problems, requires an accurate and efficient implementation of a Stekhlov-Poincaré map. In this talk, we present a perturbative technique for computing Stekhlov-Poincaré (Dirichlet to Neumann) maps for use in exterior acoustic scattering problems. This work extends the idea of using exact Dirichlet-to-Neumann maps on separable geometries. Computational experiments as presented. An error analysis for a finite element method used in conjunction with these perturbative Stekhlov-Poincaré maps is provided.

This is joint work with Prof. D. P. Nicholls, U. Notre Dame.

FRANCESCA RAPETTI, Univ. of Nice and Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
Coupling scalar and vector potentials on nonmatching grids for eddy currents in a moving conductor

The T-W formulation of the magnetic field has been introduced in many papers for the approximation of the magnetic quantities modeled by the eddy current equations. This decomposition allows to use a scalar function in the main part of the computational domain, reducing the use of vector quantities to the conducting parts. We propose to approximate these two quantities on nonmatching grids so as to be able to tackle a problem where the conducting part can move in the global domain. The connection between the two grids is managed with mortar element techniques.

JACQUES RAPPAZ, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Finite element approximation of multi-scale elliptic problems using patches of elements

In this talk we will present a numerical method for solving elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming with the initial mesh. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the CBS constant will be presented.

ERIC SONNENDRUCKER, Université Louis Pasteur, 67084 Strasbourg Cedex, France
Méthodes numériques adaptatives pour l'équation de Vlasov

La modélisation de nombreux problèmes mettant en jeu des particules chargées est basée sur l'équation de Vlasov qui est une équation de transport non linéaire dans l'espace des phases couplée aux équations de Maxwell ou de Poisson. Le fait que l'équation soit posée dans l'espace des phases double le nombre de dimensions et rend donc sa résolution numérique d'autant plus lourde. Une méthode largement utilisée pour sa résolution numérique est la méthode semi-Lagrangienne. Les physiciens l'utilisent de manière courante en 4D de l'espace des phases sur des maillages uniformes. Mais l'utilisation de maillages uniformes est loin d'être optimale et augmente de manière importante le coût de calcul. Nous présenterons dans cet exposé des méthodes adaptatives basées sur des approximations multi-résolutions permettant de manière automatique d'utiliser à chaque pas de temps un maillage parfaitement adapté à la fonction de distribution des particules qui est la quantité calculée.

JEAN-PAUL VILA, Toulouse