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CATEGORIES:Optimisation and Numerical Analysis Seminars
SUMMARY:Integral equation methods for acoustic scattering
by fractals - Dave Hewett (UCL)
DTSTART:20231114T140000Z
DTEND:20231114T150000Z
UID:TALK5359AT
URL:/talk/index/5359
DESCRIPTION:We study sound-soft time-harmonic acoustic scatter
ing by general scatterers\, including fractal scat
terers\, in 2D and 3D space. For an arbitrary comp
act scatterer Γ we reformulate the Dirichlet bound
ary value problem for the Helmholtz equation as a
first kind integral equation (IE) on Γ involving t
he Newton potential. The IE is well-posed\, except
possibly at a countable set of frequencies\, and
reduces to existing single-layer boundary IEs when
Γ is the boundary of a bounded Lipschitz open set
\, a screen\, or a multi-screen. When Γ is uniform
ly of d-dimensional Hausdorff dimension in a sense
we make precise (a d-set)\, the operator in our e
quation is an integral operator on Γ with respect
to d-dimensional Hausdorff measure\, with kernel t
he Helmholtz fundamental solution\, and we propose
a piecewise-constant Galerkin discretization of t
he IE\, which converges in the limit of vanishing
mesh width. When Γ is the fractal attractor of an
iterated function system of contracting similariti
es we prove convergence rates under assumptions on
Γ and the IE solution\, and describe a fully disc
rete implementation using recently proposed quadra
ture rules for singular integrals on fractals. We
present numerical results for a range of examples
and make our software available as a Julia code.\n
\nThis is joint work with António Caetano (Aveiro)
\, Simon Chandler-Wilde (Reading)\, Xavier Claeys
(Sorbonne)\, Andrew Gibbs (UCL) and Andrea Moiola
(Pavia).
LOCATION:Nuffield G13
CONTACT:Sergey Sergeev
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