Volume 1, Issue 6
Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation

P. Joly, J.-R. Li & S. Fliss

DOI:

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

Published online: 2006-01

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  • Abstract

We consider the solution of the Helmholtz equation −∆u(x)−n(x)2ω2u(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 Ω0 := {x ∈ Ω |a < x < a+} and that n(x) is x-periodic in Ω\Ω0. We show how to obtain exact boundary conditions on the vertical segments, Γ := {x ∈ Ω |x = a} and Γ+ := {x ∈ Ω |x = a+}, that will enable us to find the solution on Ω0 ∪Γ+ ∪Γ. Then the solution can be extended in Ω in a straightforward manner from the values on Γ and Γ+. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell. 

  • Keywords

Exact boundary conditions periodic media Dirichlet to Neumann maps.

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@Article{CiCP-1-945, author = {}, title = {Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation}, journal = {Communications in Computational Physics}, year = {2006}, volume = {1}, number = {6}, pages = {945--973}, abstract = {

We consider the solution of the Helmholtz equation −∆u(x)−n(x)2ω2u(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 Ω0 := {x ∈ Ω |a < x < a+} and that n(x) is x-periodic in Ω\Ω0. We show how to obtain exact boundary conditions on the vertical segments, Γ := {x ∈ Ω |x = a} and Γ+ := {x ∈ Ω |x = a+}, that will enable us to find the solution on Ω0 ∪Γ+ ∪Γ. Then the solution can be extended in Ω in a straightforward manner from the values on Γ and Γ+. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell. 

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7989.html} }
TY - JOUR T1 - Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation JO - Communications in Computational Physics VL - 6 SP - 945 EP - 973 PY - 2006 DA - 2006/01 SN - 1 DO - http://dor.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 -

We consider the solution of the Helmholtz equation −∆u(x)−n(x)2ω2u(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 Ω0 := {x ∈ Ω |a < x < a+} and that n(x) is x-periodic in Ω\Ω0. We show how to obtain exact boundary conditions on the vertical segments, Γ := {x ∈ Ω |x = a} and Γ+ := {x ∈ Ω |x = a+}, that will enable us to find the solution on Ω0 ∪Γ+ ∪Γ. Then the solution can be extended in Ω in a straightforward manner from the values on Γ and Γ+. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell. 

P. Joly, J.-R. Li & S. Fliss. (2020). Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation. Communications in Computational Physics. 1 (6). 945-973. doi:
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