Optimized Schwarz methods for the time-harmonic Maxwell equations with damping

In a previous paper, two of the authors have proposed and analyzed an entire hierarchy of optimized Schwarz methods for Maxwell's equations both in the time-harmonic and time-domain case. The optimization process has been perfomed in a particular situation where the electric conductivity was neglected. Here, we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case. From the mathematical point of view, the approach is different, since the algorithm does not encounter the pathological situations of the zero-conductivity case and thus the optimization problems are different. We analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones. We illustrate our analysis with numerical results

Solution of the frequency domain Maxwell equations by a high order non-conforming discontinuous Galerkin method

International audienceWe report on recent efforts towards the development of a high order, non-conforming, discontinuous Galerkin method for the solution of the system of frequency domain Maxwell's equations in heterogeneous propagation media. This method is an extension of the low order one which was proposed in (http://hal.inria.fr/inria-00155231/en/

Semaine d'Etude Mathématiques et Entreprises 5 : Reconstruction de couches géologiques à partir de données discrètes

Ce rapport synthétise le travail de recherche mis en oeuvre durant la cinquième Semaine d'Etude Maths-Entreprises à l'Ecole des Mines de Nancy. Le sujet a été proposé par le consortium GOCAD: comment reconstituer efficacement le sous-sol terrestre à partir de données discrètes éparses ? Un état de l'art est d'abord effectué sur les différentes méthodes existantes : cokrigeage statistique (Calcagno et al., 2008), discrétisation numérique (Caumon et al., 2013) et modélisation physique entre deux horizons géologiques (Hjelle and Petersen, 2011). Ensuite, nous avons tenté d'adapter l'approche (Hjelle and Petersen, 2011) à notre problématique. Il s'agit de représenter chaque couche géologique par les points d'annulation d'une fonction dont l'évolution est gérée par une loi qui contient les informations connues et permettra la reconstitution in fine du sous-sol. Finalement, on effectue la résolution numérique de l'équation de Hamilton-Jacobi associée à cette loi de propagation, s'aidant de (Osher and Fedkiw, 2003). Par souci de simplicité et surtout par manque de temps, le modèle sera résolu numériquement en 2-D et sans failles

Comparison of a one and two parameter family of transmission conditions for Maxwell's equations with damping

Transmission conditions between subdomains have a substantial influence on the convergence of iterative domain decomposition algorithms. For Maxwell's equations, transmission conditions which lead to rapidly converging algorithms have been developed both for the curl-curl formulation of Maxwell's equation, see [2, 3, 1], and also for first order formulations, see [7, 6]. These methods have well found their way into applications, see for example [9, 11, 10]. It turns out that good transmission conditions are approximations of transparent boundary conditions. For each form of approximation chosen, one can try to find the best remaining free parameters in the approximation by solving a min-max problem. Usually allowing more free parameters leads to a substantially better solution of the min-max problem, and thus to a much better algorithm. For a particular one parameter family of transmission conditions analyzed in [4], we investigate in this paper a two parameter counterpart. The analysis, which is substantially more complicated than in the one parameter case, reveals that in one particular asymptotic regime there is only negligible improvement possible using two parameters, compared to the one parameter results. This analysis settles an important open question for this family of transmission conditions, and also suggests a direction for systematically reducing the number of parameters in other optimized transmission conditions

Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell equations

We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell equations using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach does not lead at convergence of the Schwarz method to the mono-domain DG discretization, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present an alternative discretization of the transmission conditions in the framework of a DG weak formulation, and prove that for this discretization the multidomain and mono-domain solutions for the Maxwell's equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media

Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves

International audienceIt has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies

High order discontinuous Galerkin method for the solution of 2D time-harmonic Maxwell's equations

International audienceThis study is concerned with the numerical solution of 2D electromagnetic wave propagation problems in the frequency domain. We present a high order discontinuous Galerkin method formulated on unstructured triangular meshes for the solution of the system of the time-harmonic Maxwell equations in mixed form. Within each triangle, the approximation of the electromagnetic field relies on a nodal polynomial interpolation and the polynomial degree is allowed to vary across mesh elements. The resulting numerical methodology is applied to the simulation of 2D propagation problems in homogeneous and heterogeneous media as well

Approximate local magnetic-to-electric surface operators for time-harmonic Maxwell's equations

International audienceThe aim of this paper is to propose new local and accurate approximate magnetic-to-electric surface boundary operators for the three-dimensional time-harmonic Maxwell's equations. After their construction where their accuracy is improved through a regularization process, a local-ization of these operators and a full finite element approximation is introduced. Next, their numerical efficiency and accuracy is investigated in detail for different scatterers when these operators are used in the extreme situation of On-Surface Radiation Conditions methods