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

• Published in: Numerische Mathematik Vol. 106; no. 3; pp. 471 - 510
• Main Authors: DOMINGUEZ, V, GRAHAM, I. G, SMYSHLYAEV, V. P
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.05.2007; Berlin Springer Nature B.V; New York, NY
• Subjects: Acoustic scattering; Amplitudes; Asymptotic methods; Asymptotic series; Boundary integral method; Coercivity; Degrees of freedom; Domains; Exact sciences and technology; Fourier analysis; Integral equations; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Numerical methods; Plane waves; Polynomials; Robustness (mathematics); Scattering; Sciences and techniques of general use; Sequences, series, summability; Two dimensional analysis; Wavelengths; Polynomial; Freedom degree; Numerical integration; Error estimation; Acoustic wave; Fourier analysis; Boundary integral method; Acoustics; Numerical method; Bilinear form; Numerical approximation; Direct method; Sphere; Numerical analysis; Asymptotic expansion; Numerical solution; Integral equation; Relative error; Galerkin method; Asymptotic approximation; Asymptotic solution; Numerical convergence
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-007-0071-4

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 L2, with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L2-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 ∼ k1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k →∞, for a fixed number of degrees of freedom. Numerical solutions with a relative error of about 10−5 are obtained on domains of size for k up to 800 using about 60 degrees of freedom.