Publications

2025

• An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 1: general results.
  
  • Cassier Maxence
  • Joly Patrick
  , 2025. In this chapter, we investigate the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity $\varepsilon$ and magnetic permeability $\mu$ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which $\varepsilon$ and $\mu$ are rational functions of the frequency. This leads us to analyse the important classes of non-dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.
• A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic acoustic problems in heterogeneous media
  
  • Pescuma Simone
  • Gabard Gwenael
  • Chaumont-Frelet Théophile
  • Modave Axel
  Journal of Computational Physics, Elsevier, 2025, 534, pp.114009. We consider the finite element solution of time-harmonic wave propagation problems in heterogeneous media with hybridizable discontinuous Galerkin (HDG) methods. In the case of homogeneous media, it has been observed that the iterative solution of the linear system can be accelerated by hybridizing with transmission variables instead of numerical traces, as performed in standard approaches. In this work, we extend the HDG method with transmission variables, which is called the CHDG method, to the heterogeneous case with piecewise constant physical coefficients. In particular, we consider formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods are systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes. (10.1016/j.jcp.2025.114009)
  DOI : 10.1016/j.jcp.2025.114009