TY - JOUR
T1 - Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation
AU - P. Joly, J.-R. Li & S. Fliss
JO - Communications in Computational Physics
VL - 6
SP - 945
EP - 973
PY - 2006
DA - 2006/01
SN - 1
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/cicp/7989.html
KW - Exact boundary conditions
KW - periodic media
KW - Dirichlet to Neumann maps.
AB - <p style="text-align: justify;">We consider the solution of the Helmholtz equation −∆u(x)−n(x)<sup>2</sup>ω<sup>2</sup>u(x) = f(x), x = (x,y), in a domain Ω which is infinite in x and bounded in y. We assume that f(x) is supported in Ω<sup>0</sup> := {x ∈ Ω |a<sup>−</sup> &lt; x &lt; a<sup>+</sup>} and that n(x) is x-periodic in Ω\Ω<sup>0</sup>. We show how to obtain exact boundary conditions on the vertical segments, Γ<sup>−</sup> := {x ∈ Ω |x = a<sup>−</sup>} and Γ<sup>+</sup> := {x ∈ Ω |x = a<sup>+</sup>}, that will enable us to find the solution on Ω<sup>0</sup> ∪Γ<sup>+</sup> ∪Γ<sup>−</sup>. Then the solution can be extended in Ω in a straightforward manner from the values on Γ<sup>−</sup> and Γ<sup>+</sup>. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell.&nbsp;</p>