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SUMMARY:Transmission problem between periodic half-spaces - Pierre Amenoag
 badji (ENSTA ParisTech)
DTSTART:20230419T084500Z
DTEND:20230419T093000Z
UID:TALK198730@talks.cam.ac.uk
DESCRIPTION:In this talk\, we consider the 2D Helmholtz equation with coef
 ficients that coincide with different periodic functions on both sides of 
 a given interface. A numerical method has been proposed by [Fliss\, Cassan
 \, and Bernier] to solve this equation under the critical assumption that 
 the overall medium stays periodic in the direction of the interface. In fa
 ct\, in this case\, a Floquet-Bloch transform can be applied with respect 
 to the variable along the interface\, thus leading to a family of closed w
 aveguide problems. The purpose of this work is to deal with the case where
  the overall medium is no longer periodic in the direction of the interfac
 e (that is for instance if one of the half-spaces is not cut in a directio
 n of periodicity\, or if both half-spaces are periodic along the interface
 \, but with incommensurate periods). As it is done in the works of [G&eacu
 te\;rard-Varet and Masmoudi] or [Blanc\, Le Bris\, and Lions]\, we use the
  crucial (but non-obvious) observation that the medium has a quasiperiodic
  structure along the interface\, namely\, it is the restriction of a highe
 r dimensional periodic structure. Accordingly\, the idea is to interpret t
 he studied PDE as the &ldquo\;restriction&rdquo\; of an augmented PDE in h
 igher dimensions\, where periodicity along the interface is recovered. Thi
 s so-called lifting approach allows one to extend the previously developed
  ideas\, but comes with the price that the augmented equation is non-ellip
 tic (in the sense of the principal part of the differential operator)\, an
 d thus more complicated to analyse and to solve numerically. Numerical res
 ults will be provided toillustrate the method.
LOCATION:Seminar Room 1\, Newton Institute
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