Fast Solvers of the Lippmann-Schwinger Equation

  • Gennadi Vainikko
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 5)


The electromagnetic and acoustic scattering problems for the Helmholtz equation in two and three dimensions are equivalent to the Lippmann-Schwinger equation which is a weakly singular volume integral equation on the support of the scatterer. We propose for the Lippmann-Schwinger equation two discretizations of the optimal accuracy order, accompanied by fast solvers of corresponding systems of linear equations. The first method is of the second order and based on simplest cubatures; the scatterer is allowed to be only piecewise smooth. The second method is of arbitrary order and is based on a fully discrete version of the collocation method with trigonometric test functions; the scatterer is assumed to be smooth on whole space ℝn and of compact support.


Arithmetical Operation Fourier Coefficient Collocation Method Helmholtz Equation Piecewise Smooth 
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  1. [1]
    Abramowitz, M. and I. A. Stegun. (1965). Handbook of Mathematical Functions,4th Printing, United States Department of Commerce.Google Scholar
  2. [2]
    Colton, D. and R. Kress. (1992). Inverse Acoustic and Electromagnetic Scattering Theory,Springer.Google Scholar
  3. [3]
    Golub, G. H. and C. F. van Loan. (1989). Matrix Computations, John Hopkins Univ. Press, Baltimore, London.Google Scholar
  4. [4]
    Kelle, O. and G. Vainikko. (1995). A fully discrete Galerkin method for integral and pseudodifferential equations on closed curves,J. for Anal. and its Appl., Vol. 14(3), (pages 593-622).Google Scholar
  5. [5]
    Kirsch, A. and P. Monk. (1994). An analysis of coupling of finite element and Nyström methods in acoustic scattering, IMA J. Numer. Anal., Vol. 14, (pages 523 - 544 ).Google Scholar
  6. [6]
    Saranen, J. and G. Vainikko. (1996). Trigonometric collocation methods with product integration for boundary integral equations on closed curves,SIAM J. Numer. Anal., Vol. 33(4), (pages 1577-1596).Google Scholar
  7. [7]
    Vainikko, G. (1993). Multidimensional Weakly Singular Integral Equations, Lecture Notes in Math., Vol. 1549, Springer.Google Scholar
  8. [8]
    Vainikko, G. (1996). Periodic Integral and Pseudodifferential Equations, Helsinki University of Technology, Report C13.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Gennadi Vainikko
    • 1
  1. 1.Institute of MathematicsHelsinki University of TechnologyFinland

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