A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering

Abstract

We propose a new robust method for the computation of scattering of highfrequency acoustic plane waves by smooth convex objects in 2D. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. By exploiting the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary, which express the solution of the integral equation as a product of explicit oscillatory functions and more slowly varying unknown amplitudes. The amplitudes are approximated by polynomials (of minimum degree d) in each zone using a Galerkin scheme. We prove that the underlying bilinear form is continuous in L 2 , with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L 2 -coercive, independent of k, for all k sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to circular domains, but it also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as d → ∞ for fixed k. We also prove that, as k → ∞, d has to increase only very modestly to maintain a fixed error bound (d ∼ k 1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k → ∞, for a fixed number of degrees

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

“…Thus Theorem 1.1 provides alternative (and, as we shall see, much simpler) proofs of the coercivity results as k ! 1 of [28], and also shows that coercivity holds uniformly for all k on the circle and sphere provided we make the choice of coupling constant (1.16).…”

“…Since these approximation spaces depend on k, both the standard and the novel (due to Melenk) perturbation arguments, where the perturbation is k-dependent, apparently cannot be used to prove useful estimates for the stability and convergence of these hybrid methods. However, if the star-combined operator A k is used instead of A k;Á , then Theorem 1.1 gives the first stability and convergence proofs of the hybrid Galerkin methods of [20,28] in domains other than the circle/sphere. A natural question is then, how much more difficult is the star-combined operator A k to implement than the standard combined operator A k;Á ?…”

“…Fourier analysis given in [28] (although this latter proof shows that ı can be taken to be 0 in (4.16)). Note that the proof in Corollary 4.8 for the sphere does require one result obtained by Fourier analysis, namely, the bound (4.17); however, this upper bound is much easier to prove than the lower bound required for coercivity itself.…”

“…As a separate development, "hybrid asymptotic-numerical" boundary integral methods have attracted considerable recent attention [1,10,18,20,28,34,36]. These methods seek to incorporate the oscillatory components from GO and the GTD explicitly into the numerical method, and thus efficiently compute the highly oscillatory solutions.…”

Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems

“…Bruno et al [12] presented an approach with complexity independent of wavelength by restricting the interval over which boundary integrals are performed to small regions in the immediate vicinity of stationary points; Langdon and Chandler-Wilde [13] have shown that this approach is suitable for polygonal scatterers; Domínguez et al [14] demonstrated that, for problems of asymptotically large wavenumbers, the required number of degrees of freedom increases only with O(k 1/9 ), for a fixed error bound; Anand et al [15] extended this approach for problems of multiple scatterers.…”