Vol. 3, No. 3, 2021

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A Dirichlet-to-Neumann approach to the mathematical and numerical analysis in waveguides with periodic outlets at infinity

Sonia Fliss, Patrick Joly and Vincent Lescarret

Vol. 3 (2021), No. 3, 487–526
DOI: 10.2140/paa.2021.3.487
Abstract

We consider the time-harmonic scalar wave equation in junctions of several different periodic half-waveguides. In general this problem is not well-posed. Several papers propose radiation conditions, i.e., the prescription of the behavior of the solution at the infinities. This ensures uniqueness — except for a countable set of frequencies which correspond to the resonances — and yields existence when one is able to apply Fredholm’s alternative. This solution is called the outgoing solution. However, such radiation conditions are difficult to handle numerically. In this paper, we propose so-called transparent boundary conditions which enable us to characterize the outgoing solution. Moreover, the problem set in a bounded domain containing the junction with these transparent boundary conditions is of Fredholm type. These transparent boundary conditions are based on Dirichlet-to-Neumann operators whose construction is described in the paper. Contrary to the other approaches, the advantage of this approach is that a numerical method can be naturally derived in order to compute the outgoing solution. Numerical results illustrate and validate the method.

Keywords
Helmholtz equation, periodic media, waveguides, radiation condition, Dirichlet-to-Neumann maps
Mathematical Subject Classification
Primary: 35J05, 65N22, 78A50, 78A40, 78A45
Milestones
Received: 28 September 2020
Revised: 21 January 2021
Accepted: 23 February 2021
Published: 12 September 2021
Authors
Sonia Fliss
POEMS, CNRS, INRIA, ENSTA Paris
Institut Polytechnique de Paris
Palaiseau
France
Patrick Joly
POEMS, CNRS, INRIA, ENSTA Paris
Institut Polytechnique de Paris
Palaiseau
France
Vincent Lescarret
LSS, UMR8506 CNRS-SUPELEC-UPS11
Gif-sur-Yvette
France