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A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering. (English) Zbl 1169.76056

Summary: We propose a new robust method for the computation of scattering of high-frequency 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 \(d \rightarrow \infty\) for fixed \(k\). We also prove that, as \(k \rightarrow \infty, d\) has to increase only very modestly to maintain a fixed error bound \((d \sim k^{1/9}\) is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as \(k \rightarrow \infty\), for a fixed number of degrees of freedom. Numerical solutions with a relative error of about \(10^{- 5}\) are obtained on domains of size \(\mathcal{O}(1)\) for \(k\) up to 800 using about 60 degrees of freedom.

MSC:

76Q05 Hydro- and aero-acoustics
76M15 Boundary element methods applied to problems in fluid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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