Numerische Mathematik

, Volume 63, Issue 1, pp 123–144 | Cite as

Galerkin-wavelet methods for two-point boundary value problems

  • Jin-Chao Xu
  • Wei-Chang Shann
Article

Summary

Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.

Mathematics Subject Classification (1991)

65N30 65N13 65F10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson O., Lindskog G. (1986): On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math.48, 499–523Google Scholar
  2. 2.
    Calderón, A.P. (1964): Intermediate spaces and interpolation, the complex method. Studia Math.24, 113–190Google Scholar
  3. 3.
    Ciarlet, P.G. (1978): The finite element methods for elliptic problems. North-Holland AmsterdamGoogle Scholar
  4. 4.
    Coifman, R., Weiss, G. (1971): Analyse Harmonique non commutative sur certains espaces homogènes. Springer Berlin Heidelberg New YorkGoogle Scholar
  5. 5.
    Combes, J.M., Grossmann A. Tchamitchian, Ph. (eds.) (1990): Wavelets, time-frequency methods and phase space, 2nd ed. Springer, Berlin Heidelberg New YorkGoogle Scholar
  6. 6.
    Cortina, E. Gomes, S.M. (1989): A wavelet based numerical method applied to free boundary problems. IPE Technical Report, Sao Jose dos Campos, BrasilGoogle Scholar
  7. 7.
    Daubechies, I. (1988): Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math.41, 909–996Google Scholar
  8. 8.
    Daubechies, I., Lagarias, J.: Two-scale difference equations. I. Global regularity of solutions. II. Infinite matrix products, local regularity and fractals. AT&T Bell Laboratories, preprintGoogle Scholar
  9. 9.
    Duffin, R.J., Schaeffer, A.C. (1952): A class of nonharmonic Fourier series Trans. Am. Math. Soc.72, 341–366Google Scholar
  10. 10.
    Glowinski, R., Lawton, W.M., Ravachol, M., Tenenbaum, E. (1990): Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. In: R. Glowinsky, A. Lichnewsky, eds., Computing Methods in Applied Sciences and Engineering. SIAM, Philadelphia, pp. 55–120Google Scholar
  11. 11.
    Golub, G.H., Van Loan, C.F. (1988): Matrix computations, 2nd ed. Johns Hopkins University Press, BaltimoreGoogle Scholar
  12. 12.
    Goupillaud, P., Grossmann, A., Morlet, J. (1984/85): Cycle-octave and related transforms in seismic signal analysis. Geoexploration23, 85–102Google Scholar
  13. 13.
    Grossmann, A., Holschneider, M., Kronland-Martinet, R., Morlet, J. (1987): Detection of abrupt changes in sound signals with the help of wavelet transforms. In: Inverse problems: an interdisciplinary study. Academic Press, London Orlando, pp. 289–306Google Scholar
  14. 14.
    Grossmann, A., Morlet, J. (1984): Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math.15, 723–736Google Scholar
  15. 15.
    Haar, A. (1910): Zur Theorie der orthogonalen Funktionensysteme. Math. Ann.69, 331–371Google Scholar
  16. 16.
    Heil, C. (1990): Wavelets and frames. In: Signal processing, pt. 1, 2nd ed., Springer, Berlin Heidelberg New York, pp. 147–160Google Scholar
  17. 17.
    Kronland-Martinet, R., Morlet, J., Grossmann, A. (1992): Analysis of sound patterns through wavelet transforms. To appearGoogle Scholar
  18. 18.
    Latto, A., Tenenbaum, E. (1990): Les ondelettes á support compact et la solution numérique de l'équation de Burgers. Preprint, AWARE CambridgeGoogle Scholar
  19. 19.
    Liénard, J.S.: Speech analysis and reconstruction using short-time, elementary waveforms. LIMSI-CNRS, Orsay, FranceGoogle Scholar
  20. 20.
    Mallat, S. (1989): Multiresolution approximations and wavelet orthonormal bases ofL 2ℝ. Trans. Amer. Math. Soc.315, 69–87Google Scholar
  21. 21.
    Mallat, S. (1989): Multifrequency channel decompositions of images and wavelet models. IEEE Trans. Acoust. Speech Signal Process37, 2091–2110Google Scholar
  22. 22.
    Meyer, Y. (1985): Principe d'incertitude, bases hilbertiennes et algebres d'operateurs. Bourbaki Seminar, no. 662Google Scholar
  23. 23.
    Meyer, Y. (1990): Ondelettes et opérateurs, I. Ondelettes. Hermann, ParisGoogle Scholar
  24. 24.
    Rodet, X. (1984): Time-domain formant-wave-function synthesis. Comput. Music J.3, 9–14Google Scholar
  25. 25.
    Strang, G. (1989): Wavelets and dilation equations: A brief introduction. SIAM Review31, 614–627Google Scholar
  26. 26.
    Stromberg, J. (1983): A modified Franklin system and higher-order systems of ℝn as unconditional bases for Hardy spaces. In: Conference on harmonic analysis in honor of Antoni Zygmund. Wadsworth, Belmont, pp. 475–493Google Scholar
  27. 27.
    Yserentant, H. (1986): On the multi-level splitting of finite element spaces. Numer. Math.49, 379–412Google Scholar
  28. 28.
    Zienciewicz, O.C., Kelly, D.W., Gago, J., Babuška, I. (1982): Hierarchical finite element approaches, error estimates and adaptive refinement. In: The mathematics of finite elements and applications IV. Academic Press, London, pp. 313–346Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Jin-Chao Xu
    • 1
  • Wei-Chang Shann
    • 2
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.National Central UniversityChung-LiTaiwan, R.O.C.

Personalised recommendations