Monotone and Consistent discretization of the Monge-Ampere operator

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency

Derivation, justication and analysis of the Holland method for a model wave propagation problem

The Holland method is a modication of a classical Yee scheme introduced to deal with thin conducting wires when solving Maxwell's equations. This method can be very accurate, but it requires a careful calibration. There still does not exist any systematic recipe for this calibration but, in a previous work we have introduced an augmented Galerkin scheme adapted to the simulation of wave propagation in 2-D domains with small holes and we have shown that, for canonical situations, this numerical scheme could actually provide an automatic process for the calibration of the Holland method. In this talk we would like to describe precisely how, under symmetry assumptions, the traditional Holland scheme used for solving 3-D Maxwell's equations around thin wires actually reduces to this 2-D model situation. We will also present new theoretical results of numerical analysis on this subject

Integral equations via saddle point problems for time-harmonic Maxwell's equations

AbstractWe propose a new system of integral equations for the exterior time harmonic Maxwell's equation. This system is derived first from elementary manipulations of classical equations then by the minimization of a quadratic functional associated to incoming and outgoing electromagnetic waves. We analyze the inf–sup condition and various penalized problems related to this system. Then we prove that an iterative algorithm for the solution of the system of integral equations is convergent. Other numerical issues are also discussed

Conditions aux limites absorbantes d'ordre élevé pour l'équation des ondes 3D

On propose une généralisation des conditions aux limites absorbantes d'ordre élevé à l'équation des ondes 3D avec ou sans terme d'amortissement. On établit, pour les boîtes de calcul parallelipipediques, des conditions d'arêtes et de coin qui permettent de poser correctement le problème. Un schéma numérique et son algorithme associé est présenté. Un exemple numérique illustre la méthode

Boundary Conditions and Layer technique for the Simulation of Electromagnetic Waves above a Lossy Medium

Projet IDENTTwo inovative techniques for the simulation of the effect of a lossy dielectric half-space are derived and analysed. The first is an adapted impedance surface condition imposed at the interface to replace the dielectric medium. This condition, when coupled with Maxwell's equations yields a stable system which can be written in a variational form. It allows us to take into acccount both the polarization and the angle of incidence of the out-going waves. The second technique is a generalization of the Bérenger perfectly matched layers: the dielectric is now replaced by a short length layer in which the waves are governed by a modification of Maxwell's equations. For this model, the two reflection coefficients are the same as for the true dielectric while the damping inside the layer is enforced

Conditions aux limites absorbantes pour les equations de Maxwell en milieu stratifié

Projet IDENTLa résolution numérique des équations de Maxwell en régime transitoire dans un milieu de propagation infini nécessite l'adjonction de conditions aux limites absorbantes (CLA) sur les bords du domaine de calcul. Dans ce rapport, on décrit un schéma numérique s'appuyant sur des CLA dites d'ordre élevé étudiées dans un travail précédent et on donne une description fonctionnelle de l'algorithme en vue de son implémentation dans le code GORF du Centre d'étude de Gramat

Conditions absorbantes d'ordre eleve pour des modeles de propagation d'onde dans des domaines rectangulaires

Projet IDENTLa resolution numerique de problemes d'ondes en regime transitoire dans un milieu de propagation infini necessite l'adjonction de conditions aux limites absorbantes (CLA) sur les bords du domaine de calcul. Dans ce rapport, on presente et analyse les CLA dites d'ordre eleve pour l'equation des ondes 2D puis 3D avec ou sans terme d'amortissement, dans des boites de calcul rectangulaires. Concernant le probleme continu, l'apport original consiste principalement a etablir puis discuter l'existence de conditions de coin necessaires a la coherence mathematique du probleme. Quant a l'aspect discretisation, des schemas numeriques sont presentes qui permettent l'utilisation des CLA d'un point de vue pratique. Leur stabilite au sens de Kreiss est etablie dans le cas du semi-espace. Des exemples numeriques montrent l'efficacite de la methode

Perfectly Matched Absorbing Layers for the Paraxial Equations

A new absorbing boundary technique for the paraxial wave equations is proposed and analyzed. Numerical results show the efficiency of the method

Nonlocal Optimized Schwarz Methods for time-harmonic electromagnetics

We introduce a new domain decomposition strategy for time harmonic Maxwell's equations that is valid in the case of automatically generated subdomain partitions with possible presence of cross-points. The convergence of the algorithm is guaranteed and we present a complete analysis of the matrix form of the method. The method involves transmission matrices responsible for imposing coupling between subdomains. We discuss the choice of such matrices, their construction and the impact of this choice on the convergence of the domain decomposition algorithm. Numerical results and algorithms are provided

Conditions absorbantes d'ordre eleve pour les equations de Maxwell dans des domaines rectangulaires

Projet IDENTLa resolution numerique des equations de Maxwell en regime transitoire dans un milieu de propagation infini necessite l'adjonction de conditions aux limites absorbantes (CLA) sur les bords du domaine de calcul. Dans ce rapport, on propose d'etendre les CLA dites d'ordre eleve etablies dans un travail precedent pour l'equation des ondes scalaires, au systeme de Maxwell. Ces conditions s'appliquent pour des domaines de calcul parallelipipediques et consistent a imposer sur chaque face des CLA d'ordre eleve a chacunes des deux composantes tangentielles du champ electrique. Le long des aretes et aux coins, on impose des conditions qui sont determinees par des exigences de regularite de la solution recherchee. On montre que ces CLA se pretent bien a un traitement par differences finies. Des exemples numeriques montrent l'efficacite de la methode