Publications

2007

• A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners
  
  • Bourgeois Laurent
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2007, 345 (7), pp.385-390. We prove in this Note a stability estimate for ill-posed elliptic Cauchy problems in a domain with corners. This result completes an earlier result obtained for a smooth domain. To cite this article: L. Bourgeois, C. R. Acad. Sci. Paris, Ser. I 345 (2007). © 2007 Académie des sciences. (10.1016/j.crma.2007.09.014)
  DOI : 10.1016/j.crma.2007.09.014
• Locating an obstacle in a 3D finite depth ocean using the convex scattering support
  
  • Bourgeois Laurent
  • Chambeyron Colin
  • Kusiak Steven
  Journal of Computational and Applied Mathematics, Elsevier, 2007, 204 (2 SPEC. ISS.), pp.387-399. We consider an inverse scattering problem in a 3D homogeneous shallow ocean. Specifically, we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle. This reconstruction is obtained from the far-field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency. The technique is based on an implementation of the theory of the convex scattering support [S. Kusiak, J. Sylvester, The scattering support, Commun. Pure Appl. Math. (2003) 1525-1548]. A few examples are presented to show the feasibility of the method. © 2006 Elsevier B.V. All rights reserved. (10.1016/j.cam.2006.01.045)
  DOI : 10.1016/j.cam.2006.01.045
• Non-Spurious Spectral Like Element Methods for Maxwell's equations
  
  • Cohen Gary
  • Duruflé Marc
  Journal of Computational Mathematics -International Edition-, Global Science Press, 2007, pp.282-304. In this paper, we give the state of the art for the so called "mixed spectral elements" for Maxwell's equations. Several families of elements, such as edge elements and discontinuous Galerkin methods (DGM) are presented and discussed. In particular, we show the need of introducing some numerical dissipation terms to avoid spurious modes in these methods. Such terms are classical for DGM but their use for edge element methods is a novel approach described in this paper. Finally, numerical experiments show the fast and low-cost character of these elements.
• Integrability of Bianchi Universes in scalar tensor theory of gravitation
  
  • Perez Jérôme
  • Larena Julien
  Classical and Quantum Gravity, IOP Publishing, 2007, 24 (11), pp.2901. In this paper, we develop a method based on the analysis of the Kovalewski exponents to study the integrability of anisotropic and homogeneous Universes. The formalism is developed in scalar-tensor gravity, the general relativistic case appearing as a special case of this larger framework. Then, depending on the rationality of the Kovalewski exponents, the different models, both in the vacuum and in the presence of a barotropic matter fluid, are classified, and their integrability is discussed. (10.1088/0264-9381/24/11/008)
  DOI : 10.1088/0264-9381/24/11/008
• Spectral theory for an elastic thin plate floating on water of finite depth
  
  • Hazard Christophe
  • Meylan Michael H.
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2007, 68 (3), pp.629-647. The spectral theory for a two-dimensional elastic plate floating on water of finite depth is developed (this reduces to a floating rigid body or a fixed body under certain limits). Two spectral theories are presented based on the first-order and second-order formulations of the problem. The first-order theory is valid only for a massless plate, while the second-order theory applies for a plate with mass. The spectral theory is based on an inner product (different for the first- and second-order formulations) in which the evolution operator is self-adjoint. This allows the time-dependent solution to be expanded in the eigenfunctions of the self-adjoint operator which are nothing more than the single frequency solutions. We present results which show that the solution is the same as those found previously when the water depth is shallow, and show the effect of increasing the water depth and the plate mass. © 2007 Society for Industrial and Applied Mathematics. (10.1137/060665208)
  DOI : 10.1137/060665208
• Generalized formulations of Maxwell's equations for numerical Vlasov-Maxwell simulations
  
  • Ciarlet Patrick
  • Barthelmé Régine
  • Sonnendrücker Eric
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2007, 17 (5), pp.657--680. (10.1142/S0218202507002066)
  DOI : 10.1142/S0218202507002066
• Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellmann equations
  
  • Bokanowski Olivier
  • Zidani Hasnaa
  Journal of Scientific Computing, Springer Verlag, 2007, 30 (1), pp.1-33. We propose two new antidiffusive schemes for advection (or linear transport), one of them being a mixture of Roe's Super-Bee scheme and of the "Ultra-Bee" scheme. We show how to apply these schemes to treat time-dependent first order Hamilton-Jacobi-Bellman equations with discontinuous initial data, possibly infinitely-valued. Numerical tests are proposed, in one and two space dimensions, in order to validate the methods. (10.1007/s10915-005-9017-0)
  DOI : 10.1007/s10915-005-9017-0
• Asymptotic expansion of highly conductive thin sheets
  
  • Schmidt Kersten
  • Tordeux Sébastien
  , 2007, 7 issue 1, pp.2040011-2040012. (10.1002/pamm.200700278)
  DOI : 10.1002/pamm.200700278
• Asymptotic analysis for the solution to the Helmholtz problem in the exterior of a finite thin straight wire
  
  • Claeys Xavier
  , 2007. In this document we are interested in the solution of the Helmholtz equation with Dirichlet boundary condition in the exterior of a thin elongated body. We suppose that the geometry is well described in ellipsoidal coordinates. We propose an asymptotic analysis of this problem, using matched expansions. This leads to the construction of an approximate field with more explicit expression. The approximate field is composed of the first terms of the asymptotic expansion of the exact solution. Our study also leads to a validation of an acoustic version of the Pocklington's equation.
• Augmented Galerkin Schemes for the Numerical Solution of Scattering by Small Obstacles.
  
  • Claeys Xavier
  • Collino Francis
  , 2007. Dans le contexte de la propagation des ondes electromagnétiques, nous nous intéressons au problème de diffraction par des fils minces parfaitement conducteurs. Si l'on suppose que leur épaisseur est bien plus petite que la longueur d'onde caractéritique de l'onde incidente, il n'est pas posible de prendre en compte des fils minces sans faire face à un problème de verrouillage numérique. Le modèle de Holland, largement utilisé dans les codes différences finis, fournit une solution pragmatique à ce problème, en modifiant le schéma numérique sur quelques noeuds du maillage avoisinant les fils. Jusqu'à présent ce modèle n'a pas re\c cu de justification théorique solide, et il implique un paramètre appelé l'inductance linéique, qu'il doit être choisi suivant des considértions heuristiques. Nous nous intéressons ici au problème modèle de la diffraction acourtique par un petit obstacle, avec condition de Dirichlet au bord, en deux dimensions dans un milieu homogène. Nous présentons et analysons un schéma numérique qui est compatible avec les méthodes éléments finis standards (sans raffinement de maillage) et ne souffre de verrouillage numérique. Ce schéma mélange des techniques d'analyse asymptotique avec une formulation de type domaine fictif. Suivant les résultats que nous démontrons sur ce schéma, nous aboutissons à une généralisation du modèle de Holland et à un calcul automatique de l'inductance linéique. Notre analyse amène, à notre connaissance, à la première justification théorique de ce type de modèle.
• Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries
  
  • Ciarlet Patrick
  • Jamelot Erell
  Journal of Computational Physics, Elsevier, 2007, 226 (1), pp.1122-1135. A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D geometries with the help of a continuous approximation of the electromagnetic field. In this paper, we investigate how their framework can be adapted to compute the solution to the time-dependent Maxwell equations. In addition, we propose some extensions, such as the introduction of a mixed variational setting and its discretization, to handle the constraint on the divergence of the field. © 2007 Elsevier Inc. All rights reserved. (10.1016/j.jcp.2007.05.029)
  DOI : 10.1016/j.jcp.2007.05.029
• Time-harmonic acoustic propagation in the presence of a shear flow
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Duclairoir Eve-Marie
  • Legendre Guillaume
  • Mercier Jean-François
  Journal of Computational and Applied Mathematics, Elsevier, 2007, 204 (2 SPEC. ISS.), pp.428-439. This work deals with the numerical simulation, by means of a finite element method, of the time-harmonic propagation of acoustic waves in a moving fluid, using the Galbrun equation instead of the classical linearized Euler equations. This work extends a previous study in the case of a uniform flow to the case of a shear flow. The additional difficulty comes from the interaction between the propagation of acoustic waves and the convection of vortices by the fluid. We have developed a numerical method based on the regularization of the equation which takes these two phenomena into account. Since it leads to a partially full matrix, we use an iterative algorithm to solve the linear system. © 2006 Elsevier B.V. All rights reserved. (10.1016/j.cam.2006.02.048)
  DOI : 10.1016/j.cam.2006.02.048
• Construction et analyse mathématique d'un modèle approché pour la propagation d'ondes acoustiques dans un tuyau mince parcouru par un fluide en écoulement.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Duruflé Marc
  • Joly Patrick
  , 2007. Dans ce travail, nous établissons formellement un modèle aymptotique basse fréquence pour la propagation du son dans un tube mince parcouru par un fluide en écoulement stationnaire. On aboutit à une équation différentielle (de type hyperbolique) par rapport à la variable longitudinale et non local en la variable transverse. Nous menons une analyse mathématique complète du problème d'évolution correspondant. Nous étudions en particulier l'influence du profil de la vitesse de l'écoulement sur le caractère bien posé de ce problème. Cette analyse est fondée sur l'analyse spectrale d'un opérateur borné non normal en dimension infinie. Un des débouchés potentiels de ce travail est l'obtention de résultats nouveaux sur les instabilités hydrodynamiques dans le cas des fluides compressibles.
• Augmented Galerkin Schemes for the Numerical Solution of Scattering by Small Obstacles
  
  • Claeys Xavier
  • Collino Francis
  Numerische Mathematik, Springer Verlag, 2007, 116 (2), pp.243-268. We are interested in the problem of a bidimensional acoustic wave propagation in a medium including a small obstacle with homogeneous Dirichlet boundary condition. We present and analyse a numerical scheme suitable for finite elements that does not suffer from numerical locking, and takes the presence of the small obstacle into account. It is based on the fictitious domain method combined with matched asymptotic expansions. (10.1007/s00211-010-0301-z)
  DOI : 10.1007/s00211-010-0301-z
• Acoustic propagation in a flow: numerical simulation of the time-harmonic regime.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Duclairoir Eve-Marie
  • Mercier Jean-François
  ESAIM: Proceedings, EDP Sciences, 2007, 22, pp.1-14. We consider the time-harmonic acoustic radiation of a source in a moving fluid. The problem is set in a two-dimensional infinite duct and the mean flow is a subsonic parallel shear flow, with a regular profile. We deal with an equation (due to Galbrun) whose unknown is the displacement perturbation. We show how to solve the problem with a finite element method by writing a "regularized" or "augmented" formulation and using Perfectly Matched Layers to select the outgoing solution. Due to the presence of a non-local term coming from the regularization, an iterative process of resolution is preferred, which converges faster for weaker shear. Some mathematical results are established in the dissipative case. Numerical illustrations are finally presented. (10.1051/proc:072201)
  DOI : 10.1051/proc:072201
• Nonreflecting Boundary Condition for Time-Dependent Multiple Scattering
  
  • Grote Marcus J.
  • Kirsch Christoph
  Journal of Computational Physics, Elsevier, 2007, 221 (1), pp.41-62. An exact nonreflecting boundary condition (NBC) is derived for the numerical solution of time-dependent multiple scattering problems in three space dimensions, where the scatterer consists of several disjoint components. Because each sub-scatterer can be enclosed by a separate artificial boundary, the computational effort is greatly reduced and becomes independent of the relative distances between the different sub-domains. In fact, the computational work due to the NBC only requires a fraction of the computational work inside @W, due to any standard finite difference or finite element method, independently of the mesh size or the desired overall accuracy. Therefore, the overall numerical scheme retains the rate of convergence of the interior scheme without increasing the complexity of the total computational work. Moreover, the extra storage required depends only on the geometry and not on the final time. Numerical examples show that the NBC for multiple scattering is as accurate as the NBC for a single convex artificial boundary [M.J. Grote, J.B. Keller, Nonreflecting boundary conditions for time-dependent scattering, J. Comput. Phys. 127(1) (1996), 52-65], while being more efficient due to the reduced size of the computational domain. (10.1016/j.jcp.2006.06.007)
  DOI : 10.1016/j.jcp.2006.06.007