Special Colloquium
Yassine Boubendir
University of Minnesota
Domain Decomposition Methods for Scattering Problems
Abstract: Domain decomposition (DD) methods have attracted considerable
attention over the last few years, as it has become apparent that they
can afford significant flexibility in numerical simulations. Indeed,
through subtle iterative procedures, DD methods can yield significant
improvement in conditioning, they can allow for seamless coupling between
optimally suited numerical schemes and they can also enable simple
parallelization strategies. The analysis of these techniques, on the other
hand, has proven to be invaluable for applications, as it has guided the
design of rapidly convergent algorithms. In this talk, I will present some
recent results in connection with the analysis and application of (both
overlapping and non-overlapping) DD schemes that focus on problems of wave
propagation. In this context, DD methods become particularly relevant as
the oscillatory nature of waves imposes stringent demands on every
numerical algorithm. On the other hand, as I shall show in the context on
non-overlapping DD methods, the analysis of these schemes for wave
problems is substantially complicated by the need for a suitable
treatment of both ``propagating'' and evanescent'' waves, whose characters
differ significantly. I shall explain the details of the difficulties
that arise in this context and I will show that their resolution can be
based on modal analyses. I will also explain how the new DD procedures
that result from these analyses can be used to couple boundary elements
to finite-element treatments of scattering problems, for the accurate
enforcement of radiation conditions at low-to-moderate frequencies. At
the other (high-frequency) extreme of the spectrum, however, classical DD
methods are of limited use, as they still require the resolution of the
wavelength. The final part of my talk will describe a new approach,
based on integral-equation formulations, that can deliver
error-controllable solutions in frequency-independent computational times.
The central ideas here are based on the use of the geometrical optics
solution, on novel localized integration strategies, on careful treatments
of shadowing transitions, and on the use of an overlapping DD technique to
account for multiple scattering effects.
Joint work with A. Bendali, O. Bruno and F. Reitich.
Wednesday January 31, 2007 at 3:00 PM in SEO 636