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.


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
Full Text: DOI


[1] Abboud, T., Nédélec, J.-C., Zhou B.: Méthode des équations intégrales pour les hautes fréquencies. C.R. Acad. Sci. Paris 318 Série I, 165–170 (1994)
[2] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972) · Zbl 0543.33001
[3] Amini S. (1990). On the choice of the coupling parameter in boundary integral formulations of the exterior acoustics problem. Appl. Anal. 35: 75–92 · Zbl 0663.35013
[4] Arden, S., Chandler-Wilde, S.N., Langdon, S.: A collocation method for high frequency scattering by convex polygons. Reading University Numerical Analysis Report 7/05, Reading, UK. J. Comp. Appl. Math. (to appear) · Zbl 1350.76040
[5] Babich, V.M., Buldyrev, V.S.: Short-wavelength Diffraction Theory. Springer, Berlin (1991)
[6] Babich V.M., Dement’ev D.B., Samokish B.A. and Smyshlyaev V.P. (2000). On evaluation of the diffraction coefficients for arbitrary ”nonsingular” directions of a smooth convex cone. SIAM J. Appl. Math. 60: 536–573 · Zbl 0992.78016
[7] Babich, V.M., Kirpichnikova, N.Y.: The Boundary-Layer Method in Diffraction Problems. Springer, Berlin (1979) · Zbl 0411.35001
[8] Bonner B.D., Graham I.G. and Smyshlyaev V.P. (2005). The computation of conical diffraction coefficients in high-frequency acoustic wave scattering. SIAM J. Numer. Anal. 43: 1202–1230 · Zbl 1104.65115
[9] Bonner, B.D.: Calculating conical diffraction coefficients. PhD thesis, University of Bath, UK (2003)
[10] Brakhage H., Werner and P. (1965). Über das Dirichletsche Aussenraumproblem für die Helmholtzsche Schwingungsgleichung. Arch. der Math. 16: 325–329 · Zbl 0132.33601
[11] Bruno O.P., Geuzaine C.A., Monro J.A., Reitich and F. (2004). Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362: 629–645 · Zbl 1073.78004
[12] Bruno, O.P., Geuzaine, C.A., Reitich, F.: On the \(\mathcal{O}(1)\) solution of multiple-scattering problems. IEEE Trans. Magn. 41, 1488–1491 (2005)
[13] Buffa, A., Sauter, S.A.: Stabilisation of the acoustic single layer potential on non-smooth domains. SIAM J. Sci. Comput. 28, 1974–1999 (2006) · Zbl 1274.65336
[14] Burton, A.J., Miller, G.F.: The application of integral methods for the numerical solution of boundary value problems. Proc. R. Soc. Lond. Ser. A 232, 201–210 (1971) · Zbl 0235.65080
[15] Buslaev, V.S.: Short-wave asymptotic behaviour in the problem of diffraction by smooth convex contours(in Russian). Trudy Mat. Inst. Steklov. 73 14–117 (1964). Abbreviated English translation: On the shortwave asymptotic limit in the problem of diffraction by convex bodies. Sov. Phys. Dokl. 7, 685–687 (1963)
[16] Buslaev, V.S.: Formulas for the short-wave asymptotic behavior in the diffraction problem by convex bodies. (in Russian) Vestnik Leningrad. University 17 (13), 5–21 (1962)
[17] Buslaev, V.S.: The asymptotic behavior of the spectral characteristics of exterior problems for the Schrödinger operator (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 39, 149–235 (1975); English translation: Math. USSR–Izv. 9 139–223 (1975) · Zbl 0311.35010
[18] Chazarain J. (1973). Construction de la paramétrix du problème mixte hyperbolique pour l’equation des ondes. C. R. Acad. Sci. Paris 276: 1213–1215 · Zbl 0253.35058
[19] Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering. Springer, New York (1998) · Zbl 0893.35138
[20] Ecevit, F.: Integral equation formulations of electromagnetic and acoustic scattering problems: convergence of multiple scattering iterations and high-frequency asymptotic expansions. PhD thesis, University of Minnesota, (2005)
[21] Filippov V.B. (1976). Rigorous justification of the shortwave asymptotic theory of diffraction in the shadow zone. J. Sov. Math. 6: 577–626 · Zbl 0375.35017
[22] Fock, V.A.: Electromagnetic Diffraction and Propagation Problems. Pergamon Press, New York (1965)
[23] Ganesh M. and Graham I.G. (2004). A high-order algorithm for obstacle scattering in three dimensions. J. Comput. Phys. 198: 211–242 · Zbl 1052.65108
[24] Ganesh, M., Langdon, S., Sloan, I.H.: Efficient evaluation of highly oscillatory acoustic scattering surface integrals. Reading University Numerical Analysis Report 6/05, Reading, UK (2005) · Zbl 1122.65025
[25] Giebermann, K.: Schnelle Summationsverfahren zur numerischen Lösung von Integralgleichungen für Streuprobleme im \(\mathbb{R}^3\) . PhD thesis, University of Karlsruhe (1997) · Zbl 0930.65130
[26] Giladi, E., Keller, J.B.: An asymptotically derived boundary element method for the Helmholtz equation. In 20th Annual Review of Progress in Applied Computational Electromagnetics, Syracuse, New York (2004)
[27] Hörmander, L.: The Analysis of Linear Differential Operators. I, Distribution Theory and Fourier Analysis. Springer, Berlin (1983) · Zbl 0521.35001
[28] Kress R. (1985). Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering. Q. J. Mech. Appl. Math. 38: 323–341 · Zbl 0559.73095
[29] Kress R. and Spassov W.T. (1983). On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation. Numer. Math. 42: 77–85 · Zbl 0534.35027
[30] Landau L.J. (2000). Bessel functions: monotonicity and bounds. J. Lond. Math. Soc. 61: 197–215 · Zbl 0948.33001
[31] Langdon, S., Chandler-Wilde, S.N.: A wavenumber independent boundary element method for an acoustic scattering problem. Isaac Newton Institute for Mathematical Sciences Preprint NI03049-1090 CPD, 2003, SIAM J. Numer. Anal. 43, 2450–2477 (2006) · Zbl 1108.76047
[32] Langdon, S., Chandler-Wilde, S. N.: Implementation of a boundary element method for high frequency scattering by convex polygons. In: Chen, K. (ed.) Proceedings of 5th UK Conference on Boundary Integral Methods, Liverpool, pp. 2–11. University of Liverpool (2005)
[33] Lebeau G. (1984). Régularité Gevrey 3 pour la diffraction. Comm. Partial Differ. Equ. 9: 1437–1494 · Zbl 0559.35019
[34] Hargé T. and Lebeau G. (1994). Diffraction par un convexe. Invent. Math. 118: 161–196 · Zbl 0831.35121
[35] Ludwig D. (1967). Uniform asymptotic expansion of the field scattered by a convex onject at high frequencies. Comm. Pure Appl. Math. 20: 103–138 · Zbl 0154.12802
[36] Morawetz C.S. and Ludwig D. (1968). An inequality for the reduced wave equation and the justification of geometrical optics. Comm. Pure Appl. Math. 21: 187–203 · Zbl 0157.18701
[37] Melrose R.B. and Taylor M.E. (1985). Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv. Math. 55: 242–315 · Zbl 0591.58034
[38] Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Springer, New York (2001)
[39] Popov G. (1987). Some estimates of Green’s functions in the shadow. Osaka J. Math. 24: 1–12 · Zbl 0656.35022
[40] Schwab, C.:p- and hp- Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, Oxford (1998) · Zbl 0910.73003
[41] Ursell F. (1968). Creeping modes in a shadow. Proc. Camb. Philos. Soc. 68: 171–191
[42] Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1995) · Zbl 0849.33001
[43] Zayaev A.B. and Filippov V.P. (1985). Rigorous justification of the asymptotic solutions of ”sliding-wave” type. J. Sov. Math. 30: 2395–2406 · Zbl 0567.73036
[44] Zayaev A.B. and Filippov V.P. (1986). Rigorous justification of the Friedlander-Keller formulas. J. Sov. Math. 32: 134–143 · Zbl 0584.35089
[45] Zworski M. (1990). High frequency scatering by a convex obstacle. Duke Math. J. 61: 545–634 · Zbl 0732.35060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.