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A convergence theorem
for the fast multipole method
for 2 dimensional scattering problems


Author: Christophe Labreuche
Journal: Math. Comp. 67 (1998), 553-591
MSC (1991): Primary 41A58, 35J05, 65N30
DOI: https://doi.org/10.1090/S0025-5718-98-00937-5
MathSciNet review: 1458223
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Abstract: The Fast Multipole Method (FMM) designed by V. Rokhlin rapidly computes the field scattered from an obstacle. This computation consists of solving an integral equation on the boundary of the obstacle. The main result of this paper shows the convergence of the FMM for the two dimensional Helmholtz equation. Before giving the theorem, we give an overview of the main ideas of the FMM. This is done following the papers of V. Rokhlin. Nevertheless, the way we present the FMM is slightly different. The FMM is finally applied to an acoustic problem with an impedance boundary condition. The moment method is used to discretize this continuous problem.


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  • 1. M. Abramowitch, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Wiley, New York, 1964. MR 29:4914
  • 2. F. Canning, The Impedance Matrix Localization Method for Moment-Method Calculations, IEEE antenna propag.,Vol 32, pp 18-30, oct 1990.
  • 3. F. Canning, Improved Matrix Localization, IEEE antenna propag., Vol 41,No 5, pp 659-667, may 1993.
  • 4. R. Coifman, G. Beylkin, V. Rokhlin, Fast Wavelet Transforms and Numerical Algorithms, Comm. Pure Appl. Math., 44, pp 141-183, 1991. MR 92c:65061
  • 5. D. Colton, R. Kress, Integral Equation Methods in Scattering Methods in Scattering Theory, Wiley, 1983. MR 85d:35001
  • 6. L. Greengard, V. Rokhlin, A Fast Algorithm for Particle Simulations, J. Comput. Phys., Vol 73, pp 325-348, 1987. MR 88k:82007
  • 7. M.A. Hamdi, Une Formulation Variationnelle par des Équations Intégrales pour la Résolution de l'Équation de Helmholtz avec des Conditions aux Limites Mixtes, C. R. Acad. Sci. Paris, 292, série II, pp 17-20, 1981. MR 82k:35027
  • 8. A. Harten, I. Yad-Shalom, Fast Multiresolution Algorithms for Matrix-Vector Multiplication, SIAM J. Numer. Anal., Vol 31, No 4, pp 1191-1218, Aug 1994. MR 95f:65223
  • 9. A. Harten, Discrete Multi-Resolution Analysis and Generalized Wavelets, Appl. Numer. Math., Vol 12, pp 153-192, 1993. MR 95b:65163
  • 10. J. Jin, The Finite Element Method in Electromagnetics, Wiley interscience, New York, 1993.
  • 11. P. Martin, Multipole Scattering: an Invitation, in The third international conference on mathematical and numerical aspects of wave propagation, G. Cohen, 1995.
  • 12. H. Petersen, D. Soelvason, J. Perran, E. Smith, Error Estimates for the Fast Multipole Method. I. The Two-Dimensional Case, Proceedings of the Royal Society of London, serie A, Vol 448, pp 389-400, march 1995.
  • 13. V. Rokhlin, Rapid Solution of the Integral Equations of Classical Potential Theory, J. Comput. Phys., 60, pp 187-207, 1985. MR 86k:65120
  • 14. V. Rokhlin, Rapid Solution of Integral Equations of Scattering Theory in Two Dimensions, J. Comput. Phys., 86, pp 414-439, 1990. MR 90k:76081
  • 15. V. Rokhlin, B. Alpert, A Fast Algorithm for the Evaluation of Legendre Expansions, SIAM J. Sci. Stat. Comput., Vol 12, No 1, pp 158-179, Jan 1991. MR 91i:65042
  • 16. V. Rokhlin, N. Engheta, W. Murphy, M. Vassiliu, The Fast Multipole Method for Electromagnetic Scattering Problems, IEEE transactions on antenna and propagation, Vol 40, No 6, pp 634-641, june 1992. CMP 92:14
  • 17. V. Rokhlin, R. Coifman, S. Wandzura The Fast Multipole Method for Wave Equation: a Pedestrian Prescription, IEEE anten. and propag. mag., Vol 35, No 3, pp 7-12, june 1993.
  • 18. V. Rokhlin Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions, Appl. and Comput. Harmonic Analysis, Vol 1, pp 82-93, 1993. MR 95d:35034

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Additional Information

Christophe Labreuche
Affiliation: Thomson CSF-LCR, Domaine de Corbeville, 91404 Orsay cedex, France
Email: labreuch@thomson-lcr.fr

DOI: https://doi.org/10.1090/S0025-5718-98-00937-5
Keywords: Fast Multipole Method, Helmholtz equation, Hankel function
Received by editor(s): December 11, 1995
Received by editor(s) in revised form: October 7, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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