Analysis of multiple scattering iterations for high-frequency scattering problems. I: the two-dimensional case |
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Authors: | Fatih Ecevit Fernando Reitich |
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Institution: | 1.Department of Mathematics,Bo?azi?i University,Bebek, Istanbul,Turkey;2.School of Mathematics,University of Minnesota,Minneapolis,USA |
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Abstract: | We present an analysis of a recently proposed integral-equation method for the solution of high-frequency electromagnetic
and acoustic scattering problems that delivers error-controllable solutions in frequency-independent computational times.
Within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities
and on suitable extensions of the method of stationary phase. The extension to multiple-scattering configurations, in turn,
is attained through consideration of an iterative (Neumann) series that successively accounts for further geometrical wave
reflections. As we show, for a collection of two-dimensional (cylindrical) convex obstacles, this series can be rearranged
into a sum of periodic orbits (of increasing period), each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers
when these are transversed sequentially in a periodic manner. Here, we analyze the properties of these periodic orbits in
the high-frequency regime, by deriving precise asymptotic expansions for the “currents” (i.e. the normal derivative of the
fields) that they induce on the surface of the obstacles. As we demonstrate these expansions can be used to provide accurate
estimates of the rate at which their magnitude decreases with increasing number of reflections, which defines the overall
rate of convergence of the multiple-scattering series. Moreover, we show that the detailed asymptotic knowledge of these currents
can be used to accelerate this convergence and, thus, to reduce the number of iterations necessary to attain a prescribed accuracy. |
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