Numerische Mathematik

, 114:373 | Cite as

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

  • Akash Anand
  • Yassine Boubendir
  • Fatih Ecevit
  • Fernando Reitich
Article

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 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.

Mathematics Subject Classification (2000)

Primary: 65N38 Secondary: 45M05 35P25 65B99 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Akash Anand
    • 1
  • Yassine Boubendir
    • 2
  • Fatih Ecevit
    • 3
  • Fernando Reitich
    • 4
  1. 1.Applied and Computational Mathematics, CaltechPasadenaUSA
  2. 2.Department of Mathematical Sciences and Center for Applied Mathematics and StatisticsNJITNewarkUSA
  3. 3.Department of MathematicsBoğaziçi UniversityIstanbulTurkey
  4. 4.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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