Communications in Mathematical Sciences

Volume 11 (2013)

Number 4

Transparent boundary conditions for locally perturbed infinite hexagonal periodic media

Pages: 907 – 938

DOI: https://dx.doi.org/10.4310/CMS.2013.v11.n4.a2

Authors

Christophe Besse (Université Lille 1 Sciences et Technologies, Cité Scientifique, Villeneuve d’Ascq, France)

Julien Coatléven (Laboratoire Poems (UMR 7231 CNRS-INRIA-ENSTA) 32, Paris, Fance)

Sonia Fliss (Laboratoire Poems (UMR 7231 CNRS-INRIA-ENSTA) 32, Paris, Fance)

Ingrid Lacroix-Violet (Université Lille 1 Sciences et Technologies, Cité Scientifique, Villeneuve d’Ascq, France)

Karim Ramdani (Université de Lorraine, IECN, UMR 7502, Vandoeuvre-lès-Nancy, France)

Abstract

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 factoriza- tion of this operator involving two nonlocal 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 the 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 nonstandard integral equation.

Keywords

transparent boundary conditions, DtN operator, periodic media, hexagonal lattice

2010 Mathematics Subject Classification

35B27, 35J05, 35L05, 35Q40, 35Q60

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Published 15 June 2013