

Preprint 137/2006
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
Pages: 39
published in: Numerische Mathematik, 114 (2009) 2, p. 271-354
DOI number (of the published article): 10.1007/s00211-009-0249-z
Bibtex
MSC-Numbers: 65N38, 45M05, 65B99
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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.