Abstract

This work is devoted to the numerical solution of the Helmholtz equation in a 1D unbounded quasiperiodicmedium. By this, we mean that the coefficients of the model are quasi-periodic functions of the 1D spacevariable, namely the trace along a line of a periodic-function of n-variables. Except for particular choicesof the direction of this line, the resulting function is not periodic. However, the original problem can belifted onto a nD “augmented” problem with periodic coefficients : the 1D solution is the trace along thisline of the nD solution. The advantage is that the periodicity of the augmented problem allows to usethe ideas proposed for periodic Helmhotz equations Joly, Li, and Fliss, 2006. However, as the augmentedequation is degenerate (the principal part is no longer elliptic), the corresponding tools must be adaptedand new difficulties appear in both the analysis and the design of the resulting numerical method. Weshall 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. Forthe non absorbing case, we shall propose a heuristic limiting absortion procedure which will lead us toshift from the DtN to a Robin-to-Robin (RtR) method. This must be supplemented by an additionalspectral condition, in the spirit of Fliss, Joly, and Lescarret, 2021 to identify the correct physical solutionof the corresponding Riccati equation. This relies of a deep understanding of the spectral representationof the Robin propagation operator. Numerical results will be provided to illustrate the method.ReferencesFliss, Sonia, Patrick Joly, and Vincent Lescarret (2021). “A DtN approach to the mathematical andnumerical 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 waveguidescontaining a local perturbation”. In: Commun. Comput. Phys 1.6, pp. 945–973