A method for computing waveguide scattering matrices in the presence of point spectrum

Research outputpeer-review

1 Citation (Scopus)

Abstract

A waveguide G lies in ℝ n+1 , n > 1, and outside a large ball coincides with the union of finitely many non-overlapping semi-cylinders ("cylindrical ends"). The waveguide is described by the operator {L(x,D x ) - μ, B(x,D x )} of an elliptic boundary value problem in G, where L is a matrix differential operator, B is a boundary operator, and μ is a spectral parameter. The operator {L, B} is self-adjoint with respect to a Green formula. The role of L can be played, e.g., by the Helmholtz operator, by the operators in elasticity theory and hydrodynamics. As approximation for a row of the scattering matrix S(μ), we take the minimizer of a quadratic functional J R ( ,μ). To construct the functional, we solve an auxiliary boundary value problem in the bounded domain obtained by truncating the cylindrical ends of the waveguide at distance R. As R → ∞, the minimizer a(R, μ) tends with exponential rate to the corresponding row of the scattering matrix uniformly on every finite closed interval of the continuous spectrum not containing the thresholds. Such an interval may contain eigenvalues of the waveguide with eigenfunctions exponentially decaying at infinity ("trapped modes"). Eigenvalues of this sort, as a rule, occur in waveguides of complicated geometry. Therefore, in applications, the possibility to avoid worrying about (probably not detected) trapped modes turns out to be an important advantage of the method. For the reader convenience we first formulate the method for the Helmholtz operator and then present the method for the general elliptic problem.

Original languageEnglish
Title of host publicationECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers
Pages588-596
Number of pages9
Publication statusPublished - 1 Dec 2012
Event6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012 - Vienna
Duration: 10 Sep 201214 Sep 2012

Publication series

NameECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers

Conference

Conference6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012
CountryAustria
CityVienna
Period10/09/1214/09/12

Scopus subject areas

  • Computational Theory and Mathematics
  • Applied Mathematics

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