Publications

2006

• The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains
  
  • Ciarlet Patrick
  • Jung Beate
  • Kaddouri Samir
  • Labrunie Simon
  • Zou Jun
  Numerische Mathematik, Springer Verlag, 2006, 102, pp.583-610. This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used. (10.1007/s00211-005-0664-8)
  DOI : 10.1007/s00211-005-0664-8
• Une présentation mathématique de la méthode de Cagniard-de Hoop Partie II En dimension trois
  
  • Diaz Julien
  • Joly Patrick
  , 2006, pp.93. Dans ce rapport nous présentons l'extension de la méthode de Cagniard-de Hoop, que nous avons étudiée dans la première partie en dimension deux, à la dimension trois. Comme dans la première partie nous effectuons une présentation mathématique détaillée de la méthode, nous regroupons les résultats déjà connus et nous éclaircissons certaines difficultés mathématiques qui ne semblent pas avoir été traités jusqu'à maintenant .
• Un problème de Laplace non standard en milieu non borné
  
  • Tordeux Sébastien
  , 2006, pp.13. Dans le cadre des problèmes elliptiques en dimension deux, nous nous intéressons à un domaine constitué d'un demi-espace connecté à une bande infinie. Un résultat d'existence et unicité est obtenu pour un problème de Laplace inhomogène muni de comportements asymptotiques à l'infini.
• Une présentation mathématique de la méthode de Cagniard-de Hoop Partie I En dimension deux
  
  • Diaz Julien
  • Joly Patrick
  , 2006, pp.89. Dans ce rapport, nous effectuons une présentation mathématique détaillée de la méthode de Cagniard-de Hoop en dimension deux. Nous regroupons les résultats déjà connus et nous éclaircissons certaines difficultés mathématiques qui ne semblent pas avoir été traités jusqu'à maintenant .
• Influence coefficients for variational integral equations
  
  • Lenoir Marc
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2006, 343 (8), pp.561-564. We compute exact formulas for the influence coefficients deriving from the finite element discretization of integral equation methods. We consider the case of the Newtonian potential and plane triangles of the lower degree. To cite this article: M. Lenoir, C. R. Acad. Sci. Paris, Ser. I 343 (2006). © 2006 Académie des sciences. (10.1016/j.crma.2006.09.020)
  DOI : 10.1016/j.crma.2006.09.020
• Stability of Multistage Stochastic Programs
  
  • Heitsch Holger
  • Römisch Werner
  • Strugarek Cyrille
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2006, 17 (2), pp.511-525. Quantitative stability of linear multistage stochastic programs is studied. It is shown that the infima of such programs behave (locally) Lipschitz continuous with respect to the sum of an $L_{r}$‐distance and of a distance measure for the filtrations of the original and approximate stochastic (input) processes. Various issues of the result are discussed and an illustrative example is given. Consequences for the reduction of scenario trees are also discussed. Copyright © 2006 Society for Industrial and Applied Mathematics (10.1137/050632865)
  DOI : 10.1137/050632865
• Time-dependent Maxwell's equations with charges in singular geometries
  
  • Assous Franck
  • Ciarlet Patrick
  • Garcia Emmanuelle
  • Segré Jacques
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2006, 196 (1-3), pp.665-681. This paper is devoted to the solution of the instationary Maxwell equations with charges. The geometry of the domain can be singular, in the sense that its boundary can include reentrant corners or edges. The difficulties arise from the fact that those geometrical singularities generate, in their neighborhood, strong electromagnetic fields. The time-dependency of the divergence of the electric field, is addressed. To tackle this problem, some new theoretical and practical results are presented, on curl-free singular fields, and on singular fields with L2 (non-vanishing) divergence. The method, which allows to compute the instationary electromagnetic field, is based on a splitting of the spaces of solutions into a two-term direct sum. First, the subspace of regular fields: it coincides with the whole space of solutions, provided that the domain is either convex, or with a smooth boundary. Second, a singular subspace, defined and characterized via the singularities of the Laplace operator. Several numerical examples are presented, to illustrate the mathematical framework. This paper is the generalization of the singular complement method. (10.1016/j.cma.2006.07.007)
  DOI : 10.1016/j.cma.2006.07.007
• A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell's equations in time domain
  
  • Cohen Gary
  • Ferrieres Xavier
  • Pernet Sébastien
  Journal of Computational Physics, Elsevier, 2006, 217 (2), pp.340-363. In this paper, we present a non-dissipative spatial high-order discontinuous Galerkin method to solve the Maxwell equations in the time domain. The non-intuitive choice of the space of approximation and the basis functions induce an important gain for mass, stiffness and jump matrices in terms of memory. This spatial approximation, combined with a leapfrog scheme in time, leads also to a fast explicit and accurate method. A study of the dispersive error is carried out and a stability condition for the proposed scheme is established. Some comparisons with other schemes are presented to validate the new scheme and to point out its advantages. Finally, in order to improve the efficiency of the method in terms of CPU time on general unstructured meshes, a strategy of local time-stepping is proposed. (10.1016/j.jcp.2006.01.004)
  DOI : 10.1016/j.jcp.2006.01.004
• Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation
  
  • Bourgeois Laurent
  Inverse Problems, IOP Publishing, 2006, 22 (2), pp.413-430. We consider the quasi-reversibility method to solve the Cauchy problem for Laplace's equation in a smooth bounded domain. We assume that the Cauchy data are contaminated by some noise of amplitude σ, so that we make a regular choice of ε as a function of σ, where ε is the small parameter of the quasi-reversibility method. Specifically, we present two different results concerning the convergence rate of the solution of quasi-reversibility to the exact solution when σ tends to 0. The first result is a convergence rate of type 1\big/\big(\log{\frac{1}{{{\sigma}}}}\big)^\beta in a truncated domain, the second one holds when a source condition is assumed and is a convergence rate of type {{\sigma}}^{\frac{1}{2}} in the whole domain. © 2006 IOP Publishing Ltd. (10.1088/0266-5611/22/2/002)
  DOI : 10.1088/0266-5611/22/2/002
• Exact boundary conditions for periodic waveguides containing a local perturbation
  
  • Joly Patrick
  • Li Jing-Rebecca
  • Fliss Sonia
  Communications in Computational Physics, Global Science Press, 2006, 1 (6), pp.945-973. We consider the solution of the Helmholtz equation $-\Delta u({\bf x}) - n({\bf x})^2\omega^2 u({\bf x}) = f({\bf x})$, ${\bf x}=(x,y)$, in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f({\bf x})$ is supported in $\Omega^0:={{\bf x}\in {\Omega} \; | a^
• Raccordement de développements asymptotiques pour la propagation des ondes dans les milieux comportant des fentes
  
  • Joly Patrick
  • Tordeux Sébastien
  , 2006. Cet exposé porte sur la modélisation de la diffraction d'ondes en régime harmonique dans des milieux bidimensionnels comportant des fentes minces. Nous utilisons la technique des développements asymptotiques raccordés pour obtenir et justifier le développement asymptotique de la solution à tout ordre en fonction de l'épaisseur de la fente.
• Typer la dé-sérialisation sans sérialiser les types
  
  • Henry Grégoire
  • Mauny Michel
  • Chailloux Emmanuel
  , 2006, pp.133-146. In this paper, we propose a way of assigning static type information to unmarshalling functions and we describe a verification technique for unmarshalled data that preserves the execution safety provided by static type checking. This technique, whose correctness is proven, relies on singleton types whose values are transmitted to unmarshalling routines at runtime, and on an efficient checking algorithm able to deal with sharing and cycles.
• Discontinuous Galerkin methods for Maxwell's equations in the time domain
  
  • Cohen Gary
  • Ferrieres Xavier
  • Pernet Sébastien
  Comptes Rendus. Physique, Académie des sciences (Paris), 2006, 7, pp.494-500. In this article, we describe a new high-order Discontinuous Galerkin approach to Maxwell's equations in the time domain. This approach is based on hexahedral meshes and uses a mass-lumping technique. Thanks to the orthogonality of the basis functions and a judicious choice of the approximation spaces, it provides an efficient solver for these equations in terms of storage and CPU time. (10.1016/j.crhy.2006.03.004)
  DOI : 10.1016/j.crhy.2006.03.004
• Gravity, dimension, equilibrium, and thermodynamics
  
  • Perez Jérôme
  Comptes rendus de l'académie des sciences de physique, 2006, 7 (3-4), pp.406-413. It is actually possible to interpret gravitation as a property of space in a purely classical way. We note that an extended self-gravitating system equilibrium depends directly on the number of dimensions of the space in which it evolves. Given these precisions, we review the principal thermodynamical knowledge in the context of classical gravity with arbitrary dimension of space. Stability analyses for bounded 3D systems, namely the Antonov instability paradigm, are then associated to some amazing properties of globular clusters and galaxies. (10.1016/j.crhy.2006.01.011)
  DOI : 10.1016/j.crhy.2006.01.011
• High Order Generalized Impedance Boundary Conditions in Electromagnetic Scattering Problems
  
  • Duruflé Marc
  • Haddar Houssem
  • Joly Patrick
  Comptes Rendus. Physique, Académie des sciences (Paris), 2006, 7, pp.533-542. We briefly review the use and the derivation of Generalized Impedance Boundary Conditions (GIBC) in the case of thin dielectric coating and in the case of strongly absorbing medium, within the context of electromagnetic scattering problem at a fixed frequency. We then numerically test the validity and accuracy of these boundary conditions in the case of high absorption. A numerical treatment of the corner singularity is proposed to recover the accuracy of the GIBC for singular geometries.
• Perfectly matched layers for time-harmonic acoustics in the presence of a uniform flow
  
  • Bécache Eliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2006, 44 (3), pp.1191-1217. This paper is devoted to the resolution of the time-harmonic linearized Galbrun equation, which models, via a mixed Lagrangian-Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting absorption process, we characterize the outgoing solution radiated by a compactly supported source. Then we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included. © 2006 Society for Industrial and Applied Mathematics. (10.1137/040617741)
  DOI : 10.1137/040617741
• Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion
  
  • Joly Patrick
  • Tordeux Sébastien
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2006, 5 (1), pp.304--336 (electronic). In this series of two articles, we consider the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot. The Neumann boundary condition is considered along the boundary on the propagation domain, which authorizes the propagation of the wave inside the slot, even if the width of the slot is very small. We perform a complete asymptotic expansion of the solution of this problem with respect to the small parameter ε/λ, the ratio between the width of the slot, and the wavelength. We use the method of matched asymptopic expansions which allows us to describe the solution in terms of asymptotic series whose terms are characterized as the solutions of (coupled) boundary value problems posed in simple geometrical domains, independent of ε/λ: the (perturbed) half-space, the half-line, a junction zone. In this first article, we derive and analyze, from the mathematical point of view, these boundary value problems. The second one will be devoted to establishing error estimates for truncated series. (10.1137/05064494X)
  DOI : 10.1137/05064494X
• Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots
  
  • Joly Patrick
  • Tordeux Sébastien
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2006, 40 (1), pp.63--97. The authors study the asymptotic properties of the solution to the Helmoholtz equation with Neumann boundary conditions in a dumbbell-type domain in the regime when the "handle'' is thin and tightening to a curve. A mathematical analysis is done for the model problem posed in the half-plane with an infinite thin straight channel. It is proved that the solution of such a perturbed problem converges to the solution of the limiting problem for the same equation posed in the whole half-plane. Optimal estimates for the convergence rate are obtained in various norms. The authors also construct one more approximation for the solution of the perturbed problem which takes into account the presence of the channel. It is shown that this approximation is better in the sense that the estimates for the difference between this approximation and the "perturbed'' solution are smaller in order than similar estimates for the limiting solutions. The authors also conjecture that the last mentioned estimates are optimal; this conjecture is supported by a series of numerical results. (10.1051/m2an:2006008)
  DOI : 10.1051/m2an:2006008
• Etude d'un problème modèle pour la diffraction par des fils minces par développements asymptotiques raccordés Cas 2D
  
  • Claeys Xavier
  • Haddar Houssem
  • Joly Patrick
  , 2006, pp.52. Dans ce rapport, nous analysons un problème modèle pour l'étude de la diffraction d'une onde par des fils minces. Nous nous intéressons, en deux dimensions, à la solution sortante de l'équation de Helmholtz à l'extérieur d'un obstacle de petit diamètre (vis-à-vis de la longueur d'onde) sur la frontière duquel est imposée une condition de Dirichlet homogène ou une condition de Neumann homogène. Un développement à tout ordre de cette solution par rapport au diamètre de l'obstacle est obtenu.
• Computing reducing subspaces of a large linear matrix pencil
  
  • Hechme Grace
  • Nechepurenko Yuri.
  Russian Journal of Numerical Analysis and Mathematical Modelling, De Gruyter, 2006, 21 (3), pp.185-198. This paper deals with the computation of the reducing subspace associated with the rightmost part of the spectrum of a large matrix pencil A-λ B with B = diag(I,0). Two variants of the Jacobi-Davidson method are discussed and developed. One is based on the Euclidean inner product and the second on the semi-inner product induced by B. Both versions use real arithmetics and incorporate an efficient deflation procedure. Numerical results are reported. (10.1515/156939806777320359)
  DOI : 10.1515/156939806777320359
• On the convergence of the fictitious domain method for wave equation problems
  
  • Bécache Eliane
  • Rodríguez Jerónimo
  • Tsogka Chrysoula
  , 2006, pp.37. This paper deals with the convergence analysis of the fictitious domain method used for taking into account the Neumann boundary condition on the surface of a crack (or more generally an object) in the context of acoustic and elastic wave propagation. For both types of waves we consider the first order in time formulation of the problem known as mixed velocity-pressure formulation for acoustics and velocity-stress formulation for elastodynamics. The convergence analysis for the discrete problem depends on the mixed finite elements used. We consider here two families of mixed finite elements that are compatible with mass lumping. When using the first one which is less expensive and corresponds to the choice made in a previous paper, it is shown that the fictitious domain method does not always converge. For the second one a theoretical convergence analysis is presented in the acoustic case and numerical convergence is shown both for acoustic and elastic waves.