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SUMMARY:Wave propagation in unbounded quasiperiodic media - Patrick Joly (
 INRIA Saclay - Île-de-France)
DTSTART:20230419T080000Z
DTEND:20230419T084500Z
UID:TALK198727@talks.cam.ac.uk
DESCRIPTION:This work is devoted to the numerical solution of the Helmholt
 z equation in a 1D unbounded quasiperiodicmedium. By this\, we mean that t
 he coefficients of the model are quasi-periodic functions of the 1D spacev
 ariable\, namely the trace along a line of a periodic-function of n-variab
 les. Except for particular choicesof the direction of this line\, the resu
 lting function is not periodic. However\, the original problem can belifte
 d onto a nD "augmented" problem with periodic coefficients : the 1D soluti
 on is the trace along thisline of the nD solution. The advantage is that t
 he periodicity of the augmented problem allows to usethe ideas proposed fo
 r periodic Helmhotz equations Joly\, Li\, and Fliss\, 2006. However\, as t
 he augmentedequation is degenerate (the principal part is no longer ellipt
 ic)\, the corresponding tools must be adaptedand new difficulties appear i
 n both the analysis and the design of the resulting numerical method. Wesh
 all first treat the simpler case of absorbing media for which we shall dev
 elop a Dirichlet-to-Neumann(DtN) method based on a Dirichlet propagation o
 perator characterized through a Riccati equation. Forthe non absorbing cas
 e\, we shall propose a heuristic limiting absortion procedure which will l
 ead 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 r
 esults will be provided to illustrate the method.ReferencesFliss\, Sonia\,
  Patrick Joly\, and Vincent Lescarret (2021). &ldquo\;A DtN approach to th
 e mathematical andnumerical analysis in waveguides with periodic outlets a
 t infinity&rdquo\;. In: Pure and Applied Analysis.Joly\, Patrick\, Jing-Re
 becca Li\, and Sonia Fliss (2006). &ldquo\;Exact boundary conditions for p
 eriodic waveguidescontaining a local perturbation&rdquo\;. In: Commun. Com
 put. Phys 1.6\, pp. 945&ndash\;973
LOCATION:Seminar Room 1\, Newton Institute
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