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Fourier Analysis of Petrov-Galerkin Methods Based on Biorthogonal Multiresolution Analyses

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Wavelet Theory and Harmonic Analysis in Applied Sciences

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

When solving differential equations by means of a Galerkin approach, the approximating spaces are not only supposed to have good approximation properties, but also they must allow easy and fast computations. In addition, if the goal is the development of a multilevel method to detect and follow local singularities, or a multigrid scheme to solve the resulting linear systems, then hierarchical bases are necessary. As an example, we mention the finite element bases which have been widely used over the last three decades.

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References

  1. E. Bacry, S. Mallat and G. Papanicolaou. A Wavelet Based Space-Time Adaptive Numerical Mathod for Partial Differential Equations. Num. Model Math. Anal. M2 AN 26:793–834, 1992.

    MathSciNet  MATH  Google Scholar 

  2. G. Beylkin. On the representation of operators in bases of compactly supported wavelets SIAM J. Numer. Anal. 29(6): 1716–1740, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Cohen, I. Daubechies and J.-C Feauveau. Biorthogonal bases of compactly supported wavelets. Communications in Pure and Applied Mathematics, 45: 485–560, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Cortina and S. Gomes. The wavelet-Galerkin method and the KdV equation. Instituto Argentino de Matemática, preprint 218. Buenos Aires, 1994.

    Google Scholar 

  5. C. Cunha and S. Gomes. A high resolution method based on biorthogonal wavelets for the numerical simulation of conservation laws. Proceedings of the XV Congresso Ibero Latino Americano sobre Métodos Computacionais para Engenharia, Belo Horizonte, December 1994,234–243.

    Google Scholar 

  6. S. Dahlke and A. Kunoth. A biorthogonal wavelet approach for solving boundary value problems. Preprint. 1993.

    Google Scholar 

  7. S. Dahlke and I. Weinreich. Wavelet-Galerkin methods: an adapted biorthogonal wavelet basis. Constructive Approximations 9(2): 237–262, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Dahmen and C. A. Michelli. Using the refinement equation for evaluating integrals of wavelets. SIAM Jr. Numer. Anal. 30(2) 507–577, 1993.

    Article  MATH  Google Scholar 

  9. I. Daubechies. Orthonormal bases of compactly supported wavelets. Communications in Pure and applied Mathematics, 41: 909–996, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  10. I. Daubechies. “Ten Lectures on wavelets” CBMS Lecture Notes, No. 61, SIAM, Philadelphia, 1992.

    Google Scholar 

  11. R. Glowinski; W. Lawton; M. Ravachol and E. Tenembaum. Wavelet solution of Linear and Nonlinear elliptic, parabolic and hyperbolic problems in one dimension. AWARE, INC. AD890527.1, 1989.

    Google Scholar 

  12. S. Gomes and E. Cortina. Convergence estimates for the wavelet-Galerkin method. SIAM Jr. Numer. Anal. 33(1): 149–161, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Gomes, E. Cortina and I. Moroz. Characterization of biorthogonal spline wavelets by means of derivatives and primitives. In: Approximation Theory VIII. C. K. Chui, L. L. Schu-maker eds. World Scientific Publishing Co., Inc., p. 125–132, 1995.

    Google Scholar 

  14. S. Jaffard and Ph. Laurençot. Wavelets and PDEs. In: Wavelets: A tutorial in theory and Applications. Edited by Ch. Chui. Academic Press, San Diego, 1992.

    Google Scholar 

  15. A. Latto and E. Tenembaum. Compactly supported wavelets and the numerical solution of the Burgers’ equation. C.R. Acad. Sci. Pans 311 Série I: 903–909, 1990.

    MATH  Google Scholar 

  16. P.-G Lemarié, P. G. Functions a support compact dans les analyses multi-résolutions. Revista Matemática Iberoamericana 7 (2): 157–182, 1991.

    Article  MATH  Google Scholar 

  17. J. Liandrat; V. Perrier and Ph. Tchatmitchian. Numerical resolution of the regularized Burgers equation using the wavelet transform. In: Wavelets and applications, Y. Meyer ed. RMA, Masson, p 420–433, 1992.

    Google Scholar 

  18. Y. Maday, V. Perrier and J. C. Ravel. Adaptivité dynamique sur bases d’ondelettes pour l’approximation d’équations aux dérivées partielle. C. R. Acad. Sci. Pans 312 Série I: 405–410, 1991.

    MathSciNet  MATH  Google Scholar 

  19. S. Mallat. Multiresolution approximations and wavelet orthonormal bases. Trans, of the American Mathematical Society 315:69–87, 1989.

    MathSciNet  MATH  Google Scholar 

  20. Y. Meyer.“Ondelettes et Operateurs I” Hermann, Paris, 1990.

    Google Scholar 

  21. J. M. Sanz-Serna and I. Christie. Petrov-Gelerkin methods for non-linear dispersive waves. Journal of Computational Physics, 39: 94–102, 1979.

    Article  MathSciNet  Google Scholar 

  22. I. J. Schoenberg. Cardinal interpolation and spline functions. Journal of Approximation Theory, 2: 167–206, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  23. I. J. Schoenberg. Cardinal interpolation and spline functions II: interpolation of data of power growth. Journal of Approximation Theory, 6: 404–420, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. W. Schoombie. Spline Petrov-Galerkin methods for the numerical solution of the Korteveg-de Vries equation. IMA Journal of Numerical Analysis, 2: 95–109, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  25. G. Strang. Wavelets and dilation equations: a brief introduction. SIAM Review, 31 (4): 614–627, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Strang; G. A. Fix. A Fourier analysis of the finite element method. In: Constructive Aspects of Functional Analysis. Edi-zioni Cremonese, Roma, 1973.

    Google Scholar 

  27. V. Thomée. Convergence estimates for semi-discrete Galerkin methods for initial value-problems. Lecture Notes in Mathematics. 333: 243–262, 1973.

    Article  Google Scholar 

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Authors
  1. Sônia M. Gomes
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  2. Elsa Cortina
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Buenos Aires, Paseo Colón 850, 1063, Buenos Aires, Argentina

    C. E. D’Attellis

  2. Instituto de Cálculo, Ciudad Universitaria — Pabellón II, 1428, Buenos Aires, Argentina

    E. M. Fernández-Berdaguer

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© 1997 Springer Science+Business Media New York

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Gomes, S.M., Cortina, E. (1997). Fourier Analysis of Petrov-Galerkin Methods Based on Biorthogonal Multiresolution Analyses. In: D’Attellis, C.E., Fernández-Berdaguer, E.M. (eds) Wavelet Theory and Harmonic Analysis in Applied Sciences. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2010-7_6

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  • DOI: https://doi.org/10.1007/978-1-4612-2010-7_6

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7379-0

  • Online ISBN: 978-1-4612-2010-7

  • eBook Packages: Springer Book Archive

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