Commun. Comput. Phys., 1 (2006), pp. 945-973.

Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation

Patrick Joly 1*, Jing-Rebecca Li 1, Sonia Fliss 1

1 Domaine de Voluceau - Rocquencourt - B.P. 105, 78153 Le Chesnay Cedex, France.

Received 8 November 2005; Accepted (in revised version) 10 April 2006


We consider the solution of the Helmholtz equation $-\Delta u({\bf x}) - n({\bf x})^2\omega^2 u({\bf x}) = f({\bf x})$, ${\bf x}=(x,y)$, in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f({\bf x})$ is supported in $\Omega^0:={{\bf x}\in {\Omega} \; | a^-<x<a^+ }$ and that $n({\bf x})$ is $x$-periodic in ${\Omega \backslash \Omega^0}$. We show how to obtain exact boundary conditions on the vertical segments, $\Gamma^{-}:={{\bf x}\in \Omega | x=a^- }$ and $\Gamma^{+}:=\left\{{\bf x}\in \Omega \; | x=a^+ \right\}$, that will enable us to find the solution on $\Omega^0 \cup \Gamma^{+} \cup \Gamma^{-}$. Then the solution can be extended in $\Omega$ in a straightforward manner from the values on $\Gamma^{-}$ and $\Gamma^{+}$. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell.

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Key words: Exact boundary conditions; periodic media; Dirichlet to Neumann maps.

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Correspondence to: Patrick Joly , Domaine de Voluceau - Rocquencourt - B.P.~105, 78153 Le Chesnay Cedex, France. Email:

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