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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Anand A., Reitich F.: An efficient high-order algorithm for acoustic scattering from penetrable thin structures in three dimensions. J. Acoust. Soc. Am. 121, 2503–2514 (2007)
Bruno O.P., Geuzaine C.A.: An \({\mathcal{O}(1)}\) integration scheme for three-dimensional surface scattering problems. J. Comput. Appl. Math. 204(2), 463–476 (2007)
Bruno O.P., Geuzaine C.A., Monro J.A., Reitich F.: Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case. Phil. Trans. Roy. Soc. Lond. 362, 629–645 (2004)
Bruno O.P., Geuzaine C.A., Reitich F.: On the \({\mathcal{O}(1)}\) solution of multiple-scattering problems. IEEE Trans. Magn. 41, 1488–1491 (2005)
Bruno O.P., Kunyansky L.A.: A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications. J. Comput. Phys. 169(1), 80–110 (2001)
Bruno O.P., Reitich F.: High-order methods for high-frequency scattering applications. In: Ammari, H. (eds) Lecture Notes in Computational Science and Engineering vol. 59, pp. 129–164. Springer, Heidelberg (2008)
Chandler-Wilde S.N., Langdon S.: A Galerkin boundary element method for high frequency scattering by convex polygons. SIAM J. Numer. Anal. 45(2), 610–640 (2007)
Colton D., Kress R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin (1992)
Ecevit, F., Reitich, F.: A High-Frequency Integral Equation Method for Electromagnetic and Acoustic Scattering Simulations: Rate of Convergence of Multiple Scattering Iterations, Waves. Brown University, Providence, pp. 145–147 (2005)
Ecevit, F., Reitich, F.: Analysis of multiple scattering iterations for high-frequency scattering problems. I: the two-dimensional case, Numer. Math. (2009, to appear)
Ganesh M., Langdon S., Sloan I.H.: Efficient evaluation of highly oscillatory acoustic scattering surface integrals. J. Comput. Appl. Math. 204(2), 363–374 (2007)
Domínguez V., Graham I.G., Smyshlyaev V.P.: A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering. Numer. Math. 106(3), 471–510 (2007)
Hörmander L.: Fourier integral operators. I. Acta Math. 127(1–2), 79–183 (1971)
Huybrechs D., Vandewalle S.: A sparse discretization for integral equation formulations of high frequency scattering problems. SIAM J. Sci. Comput. 29(6), 2305–2328 (2007)
Melrose R.B., Taylor M.E.: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv. Math. 55, 242–315 (1985)
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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.
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Anand, A., Boubendir, Y., Ecevit, F. et al. Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case. Numer. Math. 114, 373–427 (2010). https://doi.org/10.1007/s00211-009-0263-1
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DOI: https://doi.org/10.1007/s00211-009-0263-1