# Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides

## 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.

- Authors:

- Institute of Problems of Mechanical Engineering RAS (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22773945

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 232; Journal Issue: 4; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; APPROXIMATIONS; DIRICHLET PROBLEM; FLOQUET FUNCTION; INTEGRO-DIFFERENTIAL EQUATIONS; LAPLACIAN; MATHEMATICAL SOLUTIONS; PERIODICITY; SCATTERING; WAVEGUIDES

### Citation Formats

```
Nazarov, S. A., E-mail: s.nazarov@spbu.ru, E-mail: srgnazarov@yahoo.co.uk.
```*Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3890-1.

```
Nazarov, S. A., E-mail: s.nazarov@spbu.ru, E-mail: srgnazarov@yahoo.co.uk.
```*Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides*. United States. doi:10.1007/S10958-018-3890-1.

```
Nazarov, S. A., E-mail: s.nazarov@spbu.ru, E-mail: srgnazarov@yahoo.co.uk. Sun .
"Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides". United States. doi:10.1007/S10958-018-3890-1.
```

```
@article{osti_22773945,
```

title = {Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides},

author = {Nazarov, S. A., E-mail: s.nazarov@spbu.ru, E-mail: srgnazarov@yahoo.co.uk},

abstractNote = {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.},

doi = {10.1007/S10958-018-3890-1},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 4,

volume = 232,

place = {United States},

year = {2018},

month = {7}

}