Prof. Patrick Joly | Wave propagation in unbounded quasiperiodic media

Speaker(s): Professor Patrick Joly (INRIA Saclay - Île-de-France)

Date: 19 April 2023 - 09:00 to 09:45
Venue: INI Seminar Room 1
Session Title: Wave propagation in unbounded quasiperiodic media
Event: [MWSW03] Computational methods for multiple scattering

This work is devoted to the numerical solution of the Helmholtz equation in a 1D unbounded quasiperiodic medium. By this, we mean that the coefficients of the model are quasi-periodic functions of the 1D space variable, namely the trace along a line of a periodic-function of n-variables. Except for particular choices of the direction of this line, the resulting function is not periodic. However, the original problem can be lifted onto a nD "augmented" problem with periodic coefficients : the 1D solution is the trace along this line of the nD solution. The advantage is that the periodicity of the augmented problem allows to use the ideas proposed for periodic Helmhotz equations Joly, Li, and Fliss, 2006. However, as the augmented equation is degenerate (the principal part is no longer elliptic), the corresponding tools must be adapted and new difficulties appear in both the analysis and the design of the resulting numerical method. We shall first treat the simpler case of absorbing media for which we shall develop a Dirichlet-to-Neumann (DtN) method based on a Dirichlet propagation operator characterized through a Riccati equation. For the non absorbing case, we shall propose a heuristic limiting absortion procedure which will lead us to shift from the DtN to a Robin-to-Robin (RtR) method. This must be supplemented by an additional spectral condition, in the spirit of Fliss, Joly, and Lescarret, 2021 to identify the correct physical solution of the corresponding Riccati equation. This relies of a deep understanding of the spectral representation of the Robin propagation operator. Numerical results will be provided to illustrate the method.

References Fliss, Sonia, Patrick Joly, and Vincent Lescarret (2021). “A DtN approach to the mathematical and numerical analysis in waveguides with periodic outlets at infinity”. In: Pure and Applied Analysis. Joly, Patrick, Jing-Rebecca Li, and Sonia Fliss (2006). “Exact boundary conditions for periodic waveguides containing a local perturbation”. In: Commun. Comput. Phys 1.6, pp. 945–973