Skip to main content
SpringerLink
Log in

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

We consider the numerical treatment of second kind integral equations on the real line of the form

$$\phi (s) = \psi (s) + \int_{ - \infty }^{ + \infty } {K(s - t)z(t)} \phi (t)dt, s \in \mathbb{R},$$

(abbreviatedφ =ψ +K z φ) in whichκ εL 1(ℝ),z εL ∞ (ℝ), andψ εBC(ℝ), the space of bounded continuous functions on ℝ, are assumed known andφ εBC(ℝ) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [−A, A]) via bounds on (I − K z )−1 as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on ℝ is then analysed: in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases wherez is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that ifz (related to the boundary impedance in the application) takes values in an appropriate compact subsetQ of the complex plane, then the difference betweenφ(s) and its finite section approximation computed numerically using the iterative scheme proposed is ≤C 1[khlog(1/kh)+(1−θ)−1/2(kA)−1/2] in the interval [−θA, θA] (θ<1), forkh sufficiently small, wherek is the wavenumber andh the grid spacing. Moreover this numerical approximation can be computed in ≤C 2 N logN operations, whereN = 2A/h is the number of degrees of freedom. The values of the constantsC 1 andC 2 depend only on the setQ and not on the wavenumberk or the support ofz.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amini, S., Sloan, I.H. (1989): Collocation methods for second kind integral equations with non-compact operators. J. Integral Eq. Appl.2, 1–30.

    MathSciNet  Google Scholar 

  2. Anselone, P., Sloan, I.H. (1985): Integral equations on the half-line. J. Integral Eq.9, 3–23

    MathSciNet  Google Scholar 

  3. Anselone, P., Sloan, I.H. (1988): Numerical solutions of integral equations on the half-line II. The Wiener-Hopf case. J. Integral Eq. Appl.1, 203–225

    MATH  MathSciNet  Google Scholar 

  4. Arens, T., Chandler-Wilde, S.N., Haseloh, K.O. (2001): Solvability and Spectral Properties of Integral Equations on the Real Line: I. Weighted Spaces of Continuous Functions. Submitted for publication.

  5. Atkinson, K.E. (1997): The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press

  6. Attenborough, K. (1985): Acoustical impedance models for outdoor ground surfaces. J. Sound Vib.99, 521–544

    Article  Google Scholar 

  7. Babuska, I.M., Sauter, S.A. (1997): Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers. SIAM J. Numer. Anal.34, 2392–2423

    MATH  MathSciNet  Google Scholar 

  8. Babuska, I.M., Sauter, S.A. (2000): Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers. SIAM Review42, 2451–484

    Article  MathSciNet  Google Scholar 

  9. Brakhage, H. (1960): Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode. Numer. Math.2, 183–196

    Article  MATH  MathSciNet  Google Scholar 

  10. Chan, C.H., Pak, K., Sangani, H. (1995): Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence. IEEE Trans. Anten. Prop.43, 851–859

    Article  Google Scholar 

  11. Chan, R.H., Lin, F.R. (1996): Preconditioned conjugate gradient methods for integral equations of the second kind definied on the half-line. J. Comp. Maths,14, 223–236

    MATH  MathSciNet  Google Scholar 

  12. Chan, R.H., Ng, M.K. (1996): Conjugate gradient methods for Toeplitz systems. SIAM Review,38, 427–482

    Article  MATH  MathSciNet  Google Scholar 

  13. Chandler, G.A. (1979): Product integration methods for weakly singular integral equations. Research Report, Australian National University

  14. Chandler, G.A., Graham, I.G. (1988): The convergence of Nyström methods for Wiener-Hopf equations. Numer. Math.52, 345–364

    Article  MATH  MathSciNet  Google Scholar 

  15. Chandler-Wilde, S.N. (1993): Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution. IMA J. Numer. Anal.13, 509–535

    Article  MATH  MathSciNet  Google Scholar 

  16. Chandler-Wilde, S.N. (1994): On asymptotic behavior at infinity and the finite section method for integral equations on the half-line. J. Integral Eq. Appl.6, 37–74

    MATH  MathSciNet  Google Scholar 

  17. Chandler-Wilde, S.N. (1995): Boundary value problems for the Helmholtz equation in a half-line. In: G. Cohen, ed., Proceedings of 3rd International Conference on Mathematical and Numerical Aspects of Wave Propagation, SIAM, Philadelphia

    Google Scholar 

  18. Chandler-Wilde, S.N. (1997): The impedance boundary value problem for the Helmholtz equation in a half-plane. Math. Meth. Appl. Sci.20, 813–840

    Article  MATH  MathSciNet  Google Scholar 

  19. Chandler-Wilde, S.N., Zhang, B. (1997): On the solvability of a class of second kind integral equations on unbounded domains. J. Math. Anal. Appl.214, 482–502

    Article  MATH  MathSciNet  Google Scholar 

  20. Chandler-Wilde, S.N., Zhang, B. (1998): Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate. Proc. R. Soc. Lond. A.454, 519–542

    Article  MATH  MathSciNet  Google Scholar 

  21. Chandler-Wilde, S.N., Zhang, B. (1999): Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers. SIAM J. Math. Anal.30, 559–583

    Article  MATH  MathSciNet  Google Scholar 

  22. Chandler-Wilde, S.N., Gover, M.J.C. (1989): On the application of a generalization of Toeplitz matrices to the numerical solution of integral equations with weakly singular convolution kernels. IMA J. Numerical Anal.9, 525–544

    Article  MATH  MathSciNet  Google Scholar 

  23. Chandler-Wilde S.N., Horoshenkov, K.V. (1995): Padé approximants for the acoustical characteristics of rigid frame porous media. J. Acoust. Soc. Am.98, 1119–1129

    Article  Google Scholar 

  24. Chandler-Wilde, S.N., Hothersall, D.C. (1985): Sound propagation above an inhomogeneous impedance plane. J. Sound Vib.98, 475–491

    Article  MATH  Google Scholar 

  25. Chandler-Wilde, S.N., Hothersall, D.C. (1991): On the Green function for two-dimensional acoustic propagation above a homogeneous impedance plane. Research Report, Dept. of Civil Engineering, University of Bradford, UK

    Google Scholar 

  26. Chandler-Wilde, S.N., Hothersall, D.C. (1995): Efficient calculation of the Green’s function for acoustic propagation above a homogeneous impedance place. J. Sound Vib.180, 705–724

    Article  MathSciNet  Google Scholar 

  27. Chandler-Wilde, S.N., Peplow, A.T. (1995): Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations. J. Integral Eq. Appl.7, 303–327

    MATH  MathSciNet  Google Scholar 

  28. Chandler-Wilde, S.N., Ross, C.R. (1996): Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane. Math. Meth. Appl. Sci.19, 959–976

    Article  MATH  MathSciNet  Google Scholar 

  29. Chandler-Wilde, S.N., Ross, C.R., Zhang, B. (1999): Scattering by infinite one-dimensional rough surfaces. Proc. R. Soc. Lond. A.455, 3767–3787

    MATH  MathSciNet  Google Scholar 

  30. Chandler-Wilde, S.N., Haseloh, K. Solvability and Fredholm properties of integral equations on the half-line in weighted spaces. In preparation

  31. Clemmow, P.C. (1966): The Plane Wave Spectrum Representation of Electromagnetic Fields. Pergamon, Oxford

    Google Scholar 

  32. Davis, P.J. (1979): Circulant Matrices. Wiley-Interscience

  33. Elschner, J. (1988): On spline approximation for a class of integral equations. I. Galerkin and Collocation methods with piecewise polynomials. Math. Meth. Appl. Sci.10, 543–559

    Article  MATH  MathSciNet  Google Scholar 

  34. Elschner, J. (1989): On spline collocation for convolution equations. Integral Eq. Oper. Theory.12, 486–510

    Article  MATH  MathSciNet  Google Scholar 

  35. Elschner, J. (1993): On the exponential convergence of spline approximation methods for Wiener-Hopf equations. Math. Nachr.160, 253–264

    Article  MATH  MathSciNet  Google Scholar 

  36. Gähler, S., Gähler, W. (1989): Quadrature methods for the solutions of Fredholm integral equations on the half-line. Math. Nachr.140, 321–346

    Article  MATH  MathSciNet  Google Scholar 

  37. Gohberg, I., Hanke, M., Koltracht, I. (1994): Fast preconditioned conjugate gradient algorithms for Wiener-Hopf integral equations. SIAM J. Numer. Anal.31, 429–443

    Article  MATH  MathSciNet  Google Scholar 

  38. Graham, I.G. (1982): Singularity expansions for the solutions of second kind integral equations with weakly singular convolution kernels. J. Integral Eq.4, 1–30

    MATH  Google Scholar 

  39. Habault, D. (1985): Sound propagation above an inhomogeneous plane. J. Sound Vib.100, 55–67

    Article  Google Scholar 

  40. Hackbusch, W. (1995): Integral Equations, Theory and Numerical Treatment. Birkhäuser Verlag

  41. Hackbusch, W. (1981): Die schnelle Auflösung der Fredholmschen Integralgleichung zweiter Art. Beiträge Numer. Math.9, 47–62

    Google Scholar 

  42. Hoog, F.D., Sloan, I.H. (1987): The finite-section approximation for integral equations on the half-line. J. Austral. Math. Soc. Ser B.100, 415–434

    Google Scholar 

  43. Hothersall, D.C., Chandler-Wilde, S.N. (1987): Prediction of the attenuation of road traffic noise with distance. J. Sound Vib.115, 459–472

    Article  Google Scholar 

  44. Ihlenburg, F. (1998): Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York

    MATH  Google Scholar 

  45. Kress, R. (1998): Numerical Analysis. Springer-Verlag, New York

    MATH  Google Scholar 

  46. Lin, F.R, Ng, M.K, Chan, R.H. (1997): Preconditioners for Wiener-Hopf equations with high order quadrature rules. SIAM J. Numer. Anal.34, 1418–1431

    Article  MATH  MathSciNet  Google Scholar 

  47. Oberhettinger, F., Badii, L. (1973): Tables of Laplace Transforms. Springer-Verlag

  48. Prössdorf, S., Silberman, B. (1991): Numerical analysis for integral and related operator equations. Birkhäuser Verlag

  49. Rahman, M. (1996): Numerical Treatment of a Class of Second Kind Integral Equation on the Real Line. MSc Dissertation, Dept. of Mathematics and Statistics, Brunel University

  50. Rahman, M. (2000): Fast Boundary Element Methods for Integral Equations on Infinite Domains and Scattering by Unbounded Surfaces. PhD Thesis, Dept. of Mathematical Sciences, Brunel University

  51. Roch, S., Silbermann, B. (1989): Non-strongly converging approximation methods. Demonstratiae Math.22, 651–676

    MATH  MathSciNet  Google Scholar 

  52. Ross, C.R. (1996) Direct and Inverse Scattering by Rough Surfaces. PhD Thesis, Dept. of Mathematics and Statistics, Brunel University

  53. Tsang, L., Chan, C.H., Sangani, H., Ishimaru, A., Phu, P. (1993): A banded matrix iterative approach to Monte-Carlo simulations of large-scale random rough surface scattering: TE case. J. Electromagnetic Waves Appl.7, 1185–1200

    Article  Google Scholar 

  54. Vainikko, G. (1993): Multidimensional Weakly Singular Integral Equations. Lect. Notes in Math, Springer-Verlag, Berlin

    MATH  Google Scholar 

  55. Vainikko, G. (1997): Fast Solvers of the Lippmann-Schwinger Equation. Research Reports A387, Institute of Mathematics, Helsinki University of Technology

  56. Zhang, B., Chandler-Wilde, S.N. (1998): Acoustic scattering by an inhomogeneous layer on a rigid plate. SIAM J. Appl. Math.58, 1931–1950

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Brunel University, UB8 3PH, Uxbridge, UK

    Simon N. Chandler-Wilde & Mizanur Rahman

  2. Siemens AG ICM N MC MI E71, Hofmannstrasse 51, 81359, Munich, Germany

    C. R. Ross

Authors
  1. Simon N. Chandler-Wilde
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mizanur Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. R. Ross
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by the UK Engineering and Physical Sciences Research Council and by the Radio Communications Research Unit, Rutherford Appleton Laboratory.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chandler-Wilde, S.N., Rahman, M. & Ross, C.R. A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane. Numer. Math. 93, 1–51 (2002). https://doi.org/10.1007/BF02679436

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02679436

Mathematics Subject Classification (1991)

Search

Navigation