Analysis of Multiple Scattering Iterations for High-frequency Scattering Problems. I: The Two-dimensional Case
Fatih Ecevit and Fernando Reitich
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Submission date: 24. Nov. 2006
published in: Numerische Mathematik, 114 (2009) 2, p. 271-354
DOI number (of the published article): 10.1007/s00211-009-0249-z
MSC-Numbers: 65N38, 45M05, 65B99
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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.