Section: New Results

Numerical analysis and simulation of heterogeneous systems

In optics, metamaterials (also known as negative or left-handed materials), have known a growing interest in the last two decades. These artificial composite materials exhibit the property of having negative dielectric permittivity and magnetic permeability in a certain range of frequency, leading hence to materials with negative refractive index and super lens effects. In [5] , Bunoiu and Ramdani studied a complex wave system involving such materials. More precisely, they consider a periodic homogenization problem involving two isotropic materials with conductivities of different signs: a classical material and a metamaterial (or negative material). Combining the $𝐓-$coercivity approach and the unfolding method for homogenization, they prove well-posedness results for the initial and the homogenized problems and obtain a convergence result, provided that the contrast between the two conductivities is large enough (in modulus).

Several results on domain decomposition were obtained in the frame of the collaboration of Xavier Antoine with the team of Christophe Geuzaine (Belgium). The paper [3] deals with a Schwarz-type solver for domain decomposition, the paper [8] proposes a Schwarz-type domain decomposition for high frequency electro-magnetism equations, the paper [1] exposes how to use of GPELab to solve Gross-Pitaevskii equations.

The paper [2] deals with domain decomposition for nonlinear Schrödinger equations and the book chapter [16] is focused on the modeling of Bose-Einstein condensates.