Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

We present numerical results concerning the solution of the time-harmonic Maxwell's equations discretized by discontinuous Galerkin methods. In particular, a numerical study of the convergence, which compares different strategies proposed in the literature for the elliptic Maxwell equations, is performed in the two-dimensional case.Comment: Preprint submitted for publication for the proceedings of ICCAM06 (11/04/2007

Post processing for the vector finite element method: from edge to nodal values

9 pages, preprintPurpose - Propose post processing methods for the edge finite element (FE) method on a tetrahedral mesh. They make it possible to deduce vector values on the vertices from scalar values defined on the edges of the tetrahedra. Design/methodology/approach - The new proposed techniques are based on a least squares formulation leading to a sparse matrix system to be solved. They are compared in terms of accuracy and CPU time on a FEs formulation for open boundary - frequency domain problems. Findings - A significant improvement of vector values accuracy on the vertices of the tetrahedra is obtained compared to a classical approach with a very small additional computation time. Originality/value - This work presents techniques: to obtain the values at the initial nodes of the mesh and not inside the tetrahedra; and to take into account the discontinuity to the interface between two media of different electromagnetic properties

Post processing for the vector finite element method: accurate computing of dual field

4 pagesAn accurate method to compute dual field in high frequency time harmonic problem is presented. From a primal field obtained by a vector finite element discretization, the dual field is obtained without numerical derivation by using a least square argument. The accuracy of the method is compared with the natural method using shape function derivatives

Gradient-prolongation commutativity and graph theory

This Note gives conditions that must be imposed to algebraic multilevel discretizations involving at the same time nodal and edge elements so that a gradient-prolongation commutativity condition will be satisfied; this condition is very important, since it characterizes the gradients of coarse nodal functions in the coarse edge function space. They will be expressed using graph theory and they provide techniques to compute approximation bases at each level.Comment: 6 page

Compatible Coarse Nodal and Edge Elements Through Energy Functionals

23 pagesInternational audienceWe propose new algorithms for the setup phase of algebraic multigrid AMG) solvers for linear systems coming from edge element discretization. The construction of coarse levels is performed by solving an optimization problem with a Lagrange multiplier method: we minimize the energy of coarse bases under a constraint linking coarse nodal and edge element bases. On structured meshes, the resulting AMG method and the geometric multigrid method behave similarly as preconditioners. On unstructured meshes, the convergence rate of our method compares favorably with the AMG method of Reitzinger and Schöberl

Influence of a rough thin layer on the potential

International audienceIn this paper, we study the behavior of the steadystate voltage potentials in a material composed by an interior medium surrounded by a rough thin layer and embedded in an ambient bounded medium. The roughness of the layer is supposed to be "–periodic, " being the small thickness of the layer. We present and validate numerically the rigorous approximate transmissions proved by Ciuperca et al. in [http://hal.inria.fr/inria-00356124/fr/]. This paper extends previous works in which the layer had a constant thicknes

Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

International audienceWe study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi \eps--periodic function, \eps being a small parameter, while the mean thickness of the layer is of magnitude \eps^\beta, where $\beta\in(0,1)$. Using the two-scale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor of Vogelius and Capdeboscq (ESAIM:M2AN. 2003; 37:159-173). This paper extends the previous works Poignard (Math. Meth. App. Sci. 2009; 32:435-453) and Ciuperca \textsl{et al.} (Research report INRIA RR-6812), in which $\beta\geq1$

An implicit hybridized discontinuous Galerkin method for time-domain Maxwell's equations

Discontinuous Galerkin (DG) methods have been the subject of numerous research activities in the last 15 years and have been successfully developed for various physical contexts modeled by elliptic, mixed hyperbolic-parabolic and hyperbolic systems of PDEs. One major drawback of high order DG methods is their intrinsic cost due to the very large number of globally coupled degrees of freedom as compared to classical high order conforming finite element methods. Different attempts have been made in the recent past to improve this situation and one promising strategy has been recently proposed by Cockburn (Cockburn et al., 2009) in the form of so-called hybridizable DG formulations. The distinctive feature of these methods is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements of the discretization mesh. The present work is concerned with the study of such a hybridizable DG method for the solution of the system of Maxwell equations. In this preliminary investigation, a hybridizable DG method is proposed for the two-dimensional time-domain Maxwell equations time integrated by an implicit scheme

Comparison of Methods for Modeling Uncertainties in a 2D Hyperthermia Problem

The preprint available on HAL has been submitted to IEEE Trans. on Magnetics after the conference CEFC 2008. This version has not been accepted. A revised version has been accepted to PIER and is now accessible on line at the following url: http://ceta.mit.edu/PIERB/pierb11/11.08112104.pdfInternational audienceUncertainties in biological tissue properties are weighed in the case of a hyperthermia problem. Statistic methods, experimental design and kriging technique, and stochastic methods, spectral and collocation approaches, are applied to analyze the impact of these uncertainties on the distribution of the electromagnetic power absorbed inside the body of a patient. The sensitivity and uncertainty analyses made with the different methods show that experimental designs are not suitable in this kind of problem and that the spectral stochastic method is the most efficient method only when using an adaptative algorithm

Probabilistic methods applied to 2D electromagnetic numerical dosimetry

Submitted to The international Journal for Computation and Mathematics in Electrical and Electronic Engineering (COMPEL).International audienceProbabilistic approaches are performed on electromagnetic numerical dosimetry problems in order to take into account the variability of the input parameters. These approaches are based on an expansion of the random parameters in two different ways: a spectral description and a nodal description. Compared to the Monte Carlo method, these methods are attractive since they exploit determinist codes in a more efficient way. The number of calculations can be further reduced using a regression technique, sparse grids computed from Smolyak's algorithm or a suited coordinate system. It is shown in a simple scattering problem that only 100 calculations are required applying these methods while the Monte Carlo method uses 10,000 samples