Anand, Akash and Boubendir, Yassine and Ecevit, Fatih and Reitich, Fernando (2010) Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case. Numerische Mathematik, 114 (3). pp. 373-427. ISSN 0029-599X http://resolver.caltech.edu/CaltechAUTHORS:20100129-100437703
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In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully three-dimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.
|Additional Information:||© Springer-Verlag 2009. Received: 13 August 2007. Revised: 15 July 2009. Published online: 25 September 2009. Fatih Ecevit gratefully acknowledges the support and inspiring research environment provided by the Max-Plank-Institute for Mathematics in the Sciences (Leipzig, Germany) wherein a significant part of the work presented here was performed. Fernando Reitich gratefully acknowledges support from AFOSR through contract No. FA9550-05-1-0019 and from NSF through grant No. DMS–0311763. Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant number FA9550-05-1-0019. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the US Government.|
|Classification Code:||Mathematics Subject Classification (2000) Primary: 65N38 - Secondary: 45M05 - 35P25 - 65B99.|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Tony Diaz|
|Deposited On:||01 Feb 2010 02:56|
|Last Modified:||26 Dec 2012 11:44|
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