A Sparse Discretization for Integral Equation Formulations of High Frequency Scattering Problems

Abstract

We consider two-dimensional scattering problems, formulated as an integral equation defined on the boundary of the scattering obstacle. The oscillatory nature of high-frequency scattering problems necessitates a large number of unknowns in classical boundary element methods. In addition, the corresponding discretization matrix of the integral equation is dense. We formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies. The method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns, but retains accuracy for lower frequencies. New in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals. As a result, we obtain a sparse discretization matrix for the oscillatory problem. Moreover, numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency. The sparse discretization applies to problems where the phase of the solution can be predicted a priori, for example in the case of smooth and convex scatterers.

An Efficient Method for Computing Highly Oscillatory Physical Optics Integral

“…However, it omits the higher order O (k −2 ) terms − π 2k 2 α 4 − π 2k 2 α 5 . We use three ways to compare the stationary phase point contribution results in Figure 18: Equation (35) in [18], the derived formula Equation (62) by using asymptotic approximation and the numerical SDP method …”

“…However, it generally leads to limited accuracy, especially when the object is not very large. These challenging PO type oscillatory integrals are extensively studied in [30][31][32][33][34][35][36][37]. Relevant mathematical theories and error analysis are developed to provide clearer pictures about their oscillatory behaviors.…”

Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

“…Such integrals, often referred to as Fourier-type integrals, appear in a wide area of applications, e.g., highly oscillatory scattering problems in acoustics, electromagnetics or optics [5,3,13,2]. Numerical evaluation of Fourier-type integrals with classical techniques becomes expensive as ω becomes large, which corresponds to a highly oscillatory integral.…”

A combined Filon/asymptotic quadrature method for highly oscillatory problems

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