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SUMMARY:Integral equations and boundary element methods for rough surface 
 scattering - Simon Chandler-Wilde (University of Reading)
DTSTART:20230417T100000Z
DTEND:20230417T104500Z
UID:TALK198688@talks.cam.ac.uk
DESCRIPTION:Integral equation methods are popular for the computational si
 mulation of problems of time harmonic wave scattering by (unbounded) rough
  surfaces\, but the analysis of numerical methods faces challenges\, parti
 cularly for real wavenumbers. These include but are not limited to: i) the
  possible existence of guided waves that are solutions of the homogeneous 
 problem that are localised near the rough surface (these often termed trap
 ped waves when the surface is periodic)\; ii) that the standard single- an
 d double-layer integral operators are not bounded operators on unbounded s
 urfaces (for real wavenumbers)\; iv) that standard Galerkin BEM analysis t
 echniques\, which rely on establishing that the operator is a compact pert
 urbation of a coercive operator are problematic when the surface is unboun
 ded\; v) that the infinite rough surface has to be truncated to a "finite 
 section" for practical computation.\nIn this talk we discuss probably the 
 simplest problem of this class\, acoustic scattering by a sound soft rough
  surface that is the graph of a bounded\, Lipschitz continuous function. I
 nspired by Spence et al. (Comm. Pure Appl. Math. 2011) and Chandler-Wilde 
 & Spence (arXiv:2210.02432 2022) we show that it is possible to write down
  a continuous and coercive second kind integral equation formulation for t
 his problem\, that reduces to the combined field integral equation of Chan
 dler-Wilde\, Heinemeyer\, and Potthast (Proc. R. Soc. A 2006) when the bou
 ndary is flat. This coercivity ensures stablity of every Galerkin BEM and 
 of the finite-section method\, and leads to a bound for the convergence of
  GMRES\, that the number of GMRES iterations needed to obtain a given accu
 racy is bounded independently of the mesh size and of the size of the fini
 te section. We illustrate the theory with 2D numerical results. This is jo
 int work with Martin Averseng and Euan Spence\, University of Bath.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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