Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

Akash Anand, Yassine Boubendir, Fatih Ecevit, and Fernando Reitich

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Submission date: 13. Dec. 2006
Pages: 29
published in: Numerische Mathematik, 114 (2010) 3, p. 373-427 
DOI number (of the published article): 10.1007/s00211-009-0263-1
Bibtex
MSC-Numbers: 65N38, 45M05, 65B99
Keywords and phrases: Multiple scattering, High-frequency, integral equations, Asymptotic expansions
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Abstract:
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 rigorous 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 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 ``currents'' (i.e. the normal derivative 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.