Transparent boundary conditions for locally perturbed infinite hexagonal periodic media.

Christophe Besse (1, 2) , Julien Coatleven (3, 4) , Sonia Fliss (3, 4) , Ingrid Lacroix-Violet (1, 2) , Karim Ramdani (5, 6)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 SIMPAF - SImulations and Modeling for PArticles and Fluids
3 UMA - Unité de Mathématiques Appliquées
4 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
5 CORIDA - Robust control of infinite dimensional systems and applications
6 IECL - Institut Élie Cartan de Lorraine

In this paper, we propose a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy and locally perturbed hexagonal periodic media. We obtain a factorization of this operator involving two non local operators. The first one is a DtN type operator and corresponds to a half-space problem. The second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem. The half-space DtN operator is characterized via Floquet-Bloch transform, a family of elementary strip problems and a family of stationary Riccati equations. The DtD operator is the solution of an affine operator valued equation which can be reformulated as a non standard integral equation.

Mots clés: DtN operator; periodic media; hexagonal lattice; Transparent boundary conditions

Domaines: Equations aux dérivées partielles [math.AP]

Christophe Besse, Julien Coatleven, Sonia Fliss, Ingrid Lacroix-Violet, Karim Ramdani. Transparent boundary conditions for locally perturbed infinite hexagonal periodic media.. Communications in Mathematical Sciences, 2013, 11 (4), pp.907-938. ⟨hal-00698916⟩