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A quasi-wavelet algorithm for second kind boundary integral equations

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Abstract

In solving integral equations with a logarithmic kernel, we combine the Galerkin approximation with periodic quasi-wavelet (PQW) [4]. We develop an algorithm for solving the integral equations with only O(N log N) arithmetic operations, where N is the number of knots. We also prove that the Galerkin approximation has a polynomial rate of convergence.

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References

  1. K. Amaratunga, J.R. Williams, S. Qian and J. Weiss, Wavelet-Galerkin solutions for one-dimensional partial differential equations, Internat. J. Numer. Methods Engrg. 37 (1994) 2703–2716.

    Google Scholar 

  2. G. Beylkin, R.R. Coifman and V. Rokhlin, Fast wavelet trans forms and numerical algorithms I, Commun. Pure Appl. Math. 44 (1991).

  3. M.E. Brewster and G. Beylkin, A multiresolution strategy for numerical homogenization, Appl. Comput. Anal. 2 (1995) 327–349.

    Google Scholar 

  4. H.L. Chen, Periodic orthonormal quasi-wavelet bases, Chinese Sci. Bull. 41(7) (1996).

  5. H.L. Chen and S.L. Peng, Solving integral equations with logarithmic kernel by using periodic quasi-wavelet, J. Comput. Math., to appear.

  6. Z. Chen, C.A. Micchelli and Y. Xu, The Petrov-Galerkin methods for second kind integral equations II: Multiwavelet scheme, Adv. Comput. Math. 7 (1997) 199–233.

    Google Scholar 

  7. S. Dahlke and I. Weinreich, Wavelet Galerkin methods: An adapted biorthogonal wavelet basis, Constr. Approx. 9 (1993) 237–262.

    Google Scholar 

  8. W. Dahmen, B. Kleemann, S. Prossdorf and R. Schneider, Multiscale methods for the solution of the Helmholtz and Laplace equations, Preprint (1997).

  9. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., Vol. 61 (SIAM, Philadelphia, PA, 1992).

    Google Scholar 

  10. C. de Boor and G.J. Fix, Spline approximation by quasiinterpolants, J. Approx. 8 (1973) 19–45.

    Google Scholar 

  11. D. Greenspan and P. Werner, A numerical method for the exterior Dirichlet problem for the reduced waved equation, Arch. Rational Mech. Anal. 23 (1966) 288–316.

    Google Scholar 

  12. M. Kamada, K. Toraichi and R. Mori, Periodic spline orthonormal bases, J. Approx. Theory 55 (1988) 27–34.

    Google Scholar 

  13. C.T. Kelley, A fast multilevel algorithm for integral equations, SIAM J. Numer. Anal. 32(2) (1995) 501–513.

    Google Scholar 

  14. Y.W. Koh, S.L. Lee and H.H. Tan, Periodic orthogonal splines and wavelets, Appl. Comput. Harmonic Anal. 2 (1995) 201–218.

    Google Scholar 

  15. R. Kress, Linear Integral Equations (Springer, Berlin/Heidelberg, 1989).

    Google Scholar 

  16. R. Kress and L.H. Sloan, On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation, Numer. Math. 66 (1993) 199–214.

    Google Scholar 

  17. R. Kress and W.T. Spassov, On the condition number of boundary integral operators for exterior Dirichlet problem for the Helmholtz equation, Numer. Math. 42 (1983) 77–95.

    Google Scholar 

  18. S. Mallat, Review of multifrequency channel decomposition of images and wavelet models, Technical Report 412, Robotics Report 178, New York University (1988).

  19. Y. Meyer, Principe d'incertitude, bases hilbertiennes et algèbres d'opérateurs, in: Séminaire Bourbaki 662 (Astérisque, 1985–86).

  20. C.A. Micchelli and Y. Xu, Weakly singular Fredholm integral equations I: Singularity preserving wavelet-Galerkin methods, in: Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, eds. C.K. Chui and L.L. Schumaker (1995) pp. 283–300.

  21. S.L. Peng, Wavelet transform for fast resolution of second kind integral equation with Calderón-Zygmund kernel, to appear.

  22. R. Schinzinger and P.A.A. Laura, Conformal Mapping: Methods and Applications (Elsevier, Amsterdam, 1991).

    Google Scholar 

  23. L. Schumaker, Spline Functions, Basic Theory (Wiley, New York, 1981).

    Google Scholar 

  24. Z.W. Shen and Y. Xu, Degenerate kernel schemes by wavelets for nonlinear integral equations on the real line, Appl. Anal. 59 (1995) 163–184.

    Google Scholar 

  25. M. Tasche, Orthogonal periodic spline wavelets, in: Wavelets, Image and Surface Fitting, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (1994) pp. 475–484.

  26. Y. Xu and Y. Zhao, An extrapolation method for a class of boundary integral equations, Math. Comp. 65 (1996) 587–610.

    Google Scholar 

  27. Y. Yan, A fast numerical solution for a second kind boundary integral equation with a logarithmic kernel, SIAM J. Numer. Anal. 31(2) (1994) 477–498.

    Google Scholar 

  28. Y. Yan, A fast boundary element method for the two dimensional Helmholtz equations, Comput. Methods Appl. Mech. Engrg., to appear.

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Chen, HL., Peng, SL. A quasi-wavelet algorithm for second kind boundary integral equations. Advances in Computational Mathematics 11, 355–375 (1999). https://doi.org/10.1023/A:1018992413504

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