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 123 272 F. Ecevit, F. Reitich 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.
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 threedimensional 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 123 374 A. Anand et al.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.
In this paper we develop a class of efficient Galerkin boundary element methods for the solution of two-dimensional exterior single-scattering problems. Our approach is based upon construction of Galerkin approximation spaces confined to the asymptotic behaviour of the solution through a certain direct sum of appropriate function spaces weighted by the oscillations in the incident field of radiation. Specifically, the function spaces in the illuminated/shadow regions and the shadow boundaries are simply algebraic polynomials whereas those in the transition regions are generated utilizing novel, yet simple, frequency dependent changes of variables perfectly matched with the boundary layers of the amplitude in these regions. While, on the one hand, we rigorously verify for smooth convex obstacles that these methods require only an O (k ǫ ) increase in the number of degrees of freedom to maintain any given accuracy independent of frequency, and on the other hand, remaining in the realm of smooth obstacles they are applicable in more general single-scattering configurations. The most distinctive property of our algorithms is their remarkable success in approximating the solution in the shadow region when compared with the algorithms available in the literature.
High frequency integral equation methodologies display the capability of reproducing singlescattering returns in frequency-independent computational times and employ a Neumann series formulation to handle multiple-scattering effects. This requires the solution of an enormously large number of single-scattering problems to attain a reasonable numerical accuracy in geometrically challenging configurations. Here we propose a novel and effective Krylov subspace method suitable for the use of high frequency integral equation techniques and significantly accelerates the convergence of Neumann series. We additionally complement this strategy utilizing a preconditioner based upon Kirchhoff approximations that provides a further reduction in the overall computational cost.
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