Analysis of multiple scattering iterations for high-frequency scattering problems. II

The three-dimensional scalar case

Akash Anand, Yassine Boubendir, Fatih Ecevit, Fernando Reitich

Research output: Contribution to journalArticle

16 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)373-427
Number of pages55
JournalNumerische Mathematik
Volume114
Issue number3
DOIs
StatePublished - Sep 25 2009

Fingerprint

Multiple Scattering
Multiple scattering
Scattering Problems
Scalar
Scattering
Iteration
Three-dimensional
Recurrence
Series
Rate of Convergence
Orbits
Bounce
Substructure
Periodic Orbits
Integral equations
Accelerate
Convergence Rate
Integral Equations
Continue
Orbit

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Keywords

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

Cite this

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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 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.",
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Analysis of multiple scattering iterations for high-frequency scattering problems. II : The three-dimensional scalar case. / Anand, Akash; Boubendir, Yassine; Ecevit, Fatih; Reitich, Fernando.

In: Numerische Mathematik, Vol. 114, No. 3, 25.09.2009, p. 373-427.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Analysis of multiple scattering iterations for high-frequency scattering problems. II

T2 - The three-dimensional scalar case

AU - Anand, Akash

AU - Boubendir, Yassine

AU - Ecevit, Fatih

AU - Reitich, Fernando

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