Publications

1997

• A decomposition of the electric field. Application to the Darwin model
  
  • Ciarlet Patrick
  • Sonnendrücker Eric
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 1997, 7 (8), pp.1085-1120. In many cases, the numerical resolution of Maxwell's equations is very expensive in terms of computational cost. The Darwin model, an approximation of Maxwell's equations obtained by neglecting the divergence free part of the displacement current, can be used to compute the solution more economically. However, this model requires the electric field to be decomposed into two parts for which no straightforward boundary conditions can be derived. In this paper, we consider the case of a computational domain which is not simply connected. With the help of a functional framework, a decomposition of the fields is derived. It is then used to characterize mathematically the solutions of the Darwin model on such a domain. (10.1142/S0218202597000542)
  DOI : 10.1142/S0218202597000542
• Résolution d'un cas test de contrôle optimal pour un système gouverné par l'équation des ondes à l'aide d'une méthode de décomposition de domaine
  
  • Benamou Jean-David
  , 1997. On utilise une méthode de décomposition de domaine pour résoudre un problème de contrôle optimal pour l'équation des ondes. La décomposition choisie (un sous-domaine par maille de discrétisation) permet d'utiliser facilement la technique de synthèse quadratique.
• Error Estimates for a Numerical Scheme for Ferromagnetic Problems
  
  • Monk Peter
  • Vacus Olivier
  , 1997. We propose a finite element method for approximating the non-linear equations describing the electromagnetic field in a ferromagnetic material. Using energy arguments, we prove an optimal convergence rate for the method assuming a sufficiently smooth electromagnetic field. We also verify that the finite element solution satisfies various conservation conditions appropriate to the continuous problem. Finally, we provide some numerical examples.
• A guided mode in the range of the radiation modes for a rib waveguide
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Mahé Fabrice
  Journal of Optics, Institute of Physics (IOP), 1997, 28 (1), pp.41-43. The stratified cladding of integrated optical waveguides is dispersive, which induces quite original phenomena. We present an example of a dielectric waveguide for which there exists a guided mode which propagates faster than some waves in the stratified cladding. Such a guide presents a specific symmetry and the key point is a separate study of the symmetrical and antisymmetrical modes. (10.1088/0150-536X/28/1/008)
  DOI : 10.1088/0150-536X/28/1/008
• Une caractérisation de l'orthogonal de $\Delta (H^2(\Omega)\cap H^1_0(\Omega))$ dans $L^2(\Omega)$
  
  • Assous Franck
  • Ciarlet Patrick
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 1997, 325 (6), pp.605-610. Dans le cas d'un domaine polyédrique non convexe à frontière lipschitzienne, nous donnons une caractérisation de l'orthogonal, dans L2(Ω), de l'image par le laplacien de H2(Ω) ∩ H01(Ω). Ceci permettra par la suite de décomposer la solution des équations de Maxwell en la somme d'un terme régulier et d'un terme singulier. (10.1016/S0764-4442(97)84769-6)
  DOI : 10.1016/S0764-4442(97)84769-6
• Finite element convergence for the Darwin model to Maxwell's equations
  
  • Ciarlet Patrick
  • Zou Jun
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 1997, 31 (2), pp.213-250.
• On Transitions to Stationary States in a Maxwell-Landau-Lifchitz-Gilbert System
  
  • Joly Patrick
  • Komech Alexander
  • Vacus Olivier
  , 1997. In this paper we consider Maxwell's equations together with a dissipative non-linear magnetic law, the Landau-Lifchitz-Gilbert equation, and we study long time asymptotics of solutions in the 1D case in an infinite domain of propagation. We prove long-time convergence to zero of the electroma- gnetic field in a Fréchet topology defined by local energy seminorms: this corresponds to the local decay of energy. We then introduce that the set of stationary states for the Landau-Lifchitz-Gilbert equation and prove that it corresponds to the attractor set for the distribution of magnetizationwhose presence is one of the characteristics of ferromagnetic media.
• Multivalued Solution and Viscosity Solutions of the Eikonal Equation
  
  • Benamou Jean-David
  , 1997. Some contributions to the study and computation of the multivalued solution of the Eikonal equation
• On the optimality of the median cut spectral bisection graph partitioning method
  
  • Chan Tony F.
  • Ciarlet Patrick
  • Szeto W. K.
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 1997, 18 (3), pp.943-948. Recursive spectral bisection (RSB) is a heuristic technique for finding a minimum cut graph bisection. To use this method the second eigenvector of the Laplacian of the graph is computed and from it a bisection is obtained. The most common method is to use the median of the components of the second eigenvector to induce a bisection. We prove here that this median cut method is optimal in the sense that the partition vector induced by it is the closest partition vector, in any ls norm, for $s\ge1$, to the second eigenvector. Moreover, we prove that the same result also holds for any m-partition, that is, a partition into m and (n-m)$ vertices, when using the mth largest or smallest components of the second eigenvector. Copyright © 1997 Society for Industrial and Applied Mathematics (10.1137/S1064827594262649)
  DOI : 10.1137/S1064827594262649
• A qualitative analysis of a simplified model for the non linear membrane-mallet interaction
  
  • Joly Patrick
  • Rhaouti Leila
  , 1997. In this paper, we are interested in a mathematical analysis of a simplified 1D mechanical model for the interaction between a mallet and an elastic membrane. First, using elementary mathematical techniques, we study quite precisely the case where the mallet is in interaction with a rigid body. This analysis allows us to understand the influence of physical parameters on the mallet behavior. Then, we show how the results of this study can be extended to the case where the mallet is in interaction with a membrane.