Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides
- Authors: Nazarov S.A.1,2
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Affiliations:
- Institute of Problems of Mechanical Engineering RAS
- Saint-Petersburg State University
- Issue: Vol 232, No 4 (2018)
- Pages: 461-502
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/241367
- DOI: https://doi.org/10.1007/s10958-018-3890-1
- ID: 241367
Cite item
Abstract
We consider the Dirichlet and Neumann problems for the Laplace operator in periodic waveguides. Integro-differential connections between the solution and its normal derivative, interpreted as a finite-dimensional version of the Steklov–Poincaré operator, are imposed on the artificial face of the truncated waveguide. These connections are obtained from the orthogonality and normalization conditions for the Floquet waves which are oscillating incoming/outgoing, as well as exponentially decaying/growing in the periodic waveguide. Under certain conditions, we establish the unique solvability of the problem and obtain error estimates for the solution itself, as well as for scattering coefficients in the solution. We give examples of trapped waves in periodic waveguides.
About the authors
S. A. Nazarov
Institute of Problems of Mechanical Engineering RAS; Saint-Petersburg State University
Author for correspondence.
Email: s.nazarov@spbu.ru
Russian Federation, 61, V.O., Bolshoj pr., St. Petersburg, 199178; 7-9, Universitetskaya nab., St. Petersburg, 199034