Research Directions

Our research activities are concerned with the design, analysis and high performance implementation of numerical methods for the simulation of the interaction of waves (electromagnetic waves and elastic waves) with complex media and irregularly shaped structures.

High order discretization methods

We concentrate our efforts on finite element type methods belonging to the family of Discontinuous Galerkin (DG) methods. DG methods are at the heart of the activities of the team regarding the development of high order discretization schemes for the differential systems modeling time-domain and time-harmonic electromagnetic and elastodynamic wave propagation. We currently study three variants of DG methods: (1) DG methods for time-domain problems, (2) hybridizable DG (HDG) methods for time-domain and time-harmonic problems and (3) multiscale DG methods for time-domain problems.

Efficient time integration strategies

The use of unstructured meshes in conjunction with high order DG discretization methods for time-domain problems (so-called DGTD methods) is appealing for dealing with complex geometries and heterogeneous propagation media. Moreover, DG discretization methods are naturally adapted to local, conforming as well as non-conforming, refinement of the underlying mesh. Most of the existing DGTD methods rely on explicit time integration schemes and lead to block diagonal mass matrices which is often recognized as one of the main advantages with regards to continuous finite element methods. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. In this context, we study accurate and efficient strategies combining explicit and implicit time integration schemes.

Numerical treatment of complex material models

Towards the general aim of being able to consider concrete physical situations, we are interested in taking into account in the numerical methodologies that we study, a better description of the propagation of waves in realistic media. For example, in the context of DGTD formulations for electromagnetic wave propagation models, we study the numerical treatment of local and non-local dispersion models.

Applications

Although our methodological contributions are in theory applicable to a wide panel of applications in electromagnetics and elastodynamics, we currently concentrate our efforts on devising innovative numerical methodologies for the simulation of problems involving waves interacting with matter structured at the nanoscale. As a first step in this direction we consider applications pertaining to nanophotonics. Recent achievements and sample results are described here.