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Seminario del 2018
2018
20 giugno
Andrea Moiola
Seminario di analisi numerica
The mathematical analysis and numerical simulation of acoustic and
electromagnetic wave scattering by planar screens is a classical
topic. The standard technique involves reformulating the problem as a
boundary integral equation on the screen, which can be solved
numerically using a boundary element method. Theory and computation
are both well-developed for the case where the screen is an open
subset of the plane with smooth (e.g. Lipschitz or smoother) boundary.
In this talk I will explore the case where the screen is an arbitrary
subset of the plane; in particular, the screen could have fractal
boundary, or itself be a fractal. Such problems are of interest in the
study of fractal antennas in electrical engineering, light scattering
by snowflakes/ice crystals in atmospheric physics, and in certain
diffraction problems in laser optics. The roughness of the screen
presents challenging questions concerning how boundary conditions
should be enforced, and the appropriate function space setting. But
progress is possible and there is interesting behaviour to be
discovered: for example, a sound-soft screen with zero area (planar
measure zero) can scatter waves provided the fractal dimension of the
set is large enough. This research has also motivated investigations
into the properties of fractional Sobolev spaces (the classical Bessel
potential spaces) on non-Lipschitz domains. Accurate computations are
also challenging because of the need to adapt the mesh to the fine
structure of the fractal. As well as presenting numerical results, I
will outline some outstanding open questions. This is joint work with
Simon Chandler-Wilde (Reading) and David Hewett (UCL).