TY - GEN

T1 - A method for computing waveguide scattering matrices in the presence of point spectrum

AU - Plamenevskii, Boris

AU - Sarafanov, Oleg

PY - 2012/12/1

Y1 - 2012/12/1

N2 - 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.

AB - 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.

KW - Cylindric outlets

KW - Elliptic problems

KW - Green formula

KW - Incoming and outgoing waves

KW - Selfadjoint problems

KW - Waveguide

UR - http://www.scopus.com/inward/record.url?scp=84871630309&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84871630309

SN - 9783950353709

T3 - ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers

SP - 588

EP - 596

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

T2 - 6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012

Y2 - 10 September 2012 through 14 September 2012

ER -