## Section: New Results

### Discontinuous Galerkin methods for the Maxwell equations

#### DGTD- method based on hierarchical polynomial interpolation

Participants : Loula Fezoui, Joseph Charles, Stéphane Lanteri.

In the high order DGTD- methods developed by the team so far, the local approximation of the electromagnetic field relies on a nodal (Lagrange type) polynomial interpolation method, however it is clear that other polynomial interpolation methods could be adopted as well. The choice of a set ot basis functions should ideally take into account several criteria among which, the modal or nodal nature of the functions, the orthogonality of the functions, the hierarchical structure of the functions, the conditioning of the elemental matrices to be inverted (e.g the local mass matrix in time explicit DGTD methods) and the programming simplicity. The goal of this work is to design a high order DGTD- method based on hierarchical polynomial basis expansions on simplicial elements in view of the development of a p -adaptive solution strategy. As a first step, we consider using the conforming hierarchical polynomial basis expansions described in [44] . Afterwards, we will study the possibility of relaxing some of the conformity requirements in the construction of the previous bases in order to obtain a hierarchical interpolation method which is better adapted to the DG discretization framework.

#### DGTD- method on multi-element meshes

Participants : Clément Durochat, Stéphane Lanteri, Mark Loriot [ Distene, Pôle Teratec, Bruyères-le-Chatel ] .

There exist several propagation settings for which the use of a single geometrical element type (a simplex in the DGTD methods developed by the team so far) in the computational domain discretization process may not be optimal. Instead, one would ideally allow the combined use of different element types e.g quadrangles and triangles in the 2D case, or hexahedra and tetrahedra in the 3D case, possibly in a non-conforming way (i.e allowing hanging nodes). This work is a continuation of the study initiated in 2008 which was concerned with the design of a hybrid FVTD/DGTD- method on conforming meshes consisting of quadrangular and triangular elements. We have studied this year a DGTD- method formulated on conforming hybrid quadrangular/triangular meshes [31] and relying on nodal polynomial interpolation. This is part of an ongoing effort which aims at developing a flexible DGTD- method on non-conforming hybrid hexahedral/tetrahedral meshes for the numerical simulation of 3D time domain electromagnetic wave propagation problems.

#### DGTD- method for dispersive materials

Participants : Claire Scheid, Loula Fezoui, Maciej Klemm [ Electromagnetics Group, University of Bristol, UK ] , Stéphane Lanteri.

A medium is called dispersive if the speed of the wave that propagates in this medium depends on the frequency. There exists different physical models of dispersion whose characteristics mainly depend on the considered medium. Two main strategies can be considered for the numerical treatment of a model characterizing a dispersive material: the recursive convolution method (RC) and the auxiliary differential equation method (ADE) [45] . We have initiated this year a study aiming at the development of a numerical methodology combining a high order DGTD- method on triangular meshes with an auxiliary differential equation modeling the time evolution of the electric polarization for a dispersive medium of Debye type. This work comprises both theoretical aspects (stability and convergence analysis of the resulting DGTD- method for the time domain Maxwell equations for dispersive media, and application aspects. In particular, a collaboration has been initiated with the Electromagnetics Group of the University of Bristol which is designing a radar-based imaging system for breast tumors.

#### DGFD- method for the frequency domain Maxwell equations

Participants : Victorita Dolean, Mohamed El Bouajaji, Stéphane Lanteri, Ronan Perrussel [ Ampère Laboratory, Ecole Centrale de Lyon ] .

A large number of electromagnetic wave propagation problems can be modeled by assuming a time harmonic behavior and thus considering the numerical solution of the time harmonic (or frequency domain) Maxwell equations. In this study, we investigate the applicability of discontinuous Galerkin methods on simplicial meshes for the calculation of time harmonic electromagnetic wave propagation in heterogeneous media. Although there are clear advantages of using DG methods based on a centered scheme for the evaluation of surface integrals when solving time domain problems [8] , such a choice is questionable in the context of time harmonic problems. Penalized DG formulations or DG formulations based on an upwind numerical flux have been shown to yield optimally convergent high order DG methods [38] . Moreover, such formulations are necessary to prevent the apparition of spurious modes when solving the Maxwell eigenvalue problem [49] . We have developed this year an arbitrary high order discontinuous Galerkin frequency domain DGFD- method on triangular meshes, relying on either a centered or an upwind flux, for solving the 2D time harmonic Maxwell equations [17] (invited oral presentation at the Compumag 2009 conference). Moreover, as a first step towards the development of a p -adaptive DGFD- method, the approximation order is allowed to be defined at the element level based on a local geometrical criterion.