Abstract
The aim of this paper is to provide a large class of scaling functions for which the convergence analysis for the Galerkin method developed in [9] is applicable, whereas in that paper the only scaling functions considered for practical applications are B-splines and a few of the orthonormal Daubechies scaling functions. The functions considered here, were recently introduced in [12] where it was proved that they satisfy many properties making them interesting for the applications. In particular, here we show that the use of these functions has some advantages with respect to other basis functions.
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C. Canuto, A. Tabacco and K. Urban, The wavelet element method. Part I: Construction and analysis, Appl. Comput. Harmon. Anal. 6 (1999) 1–52.
A. Cohen, Numerical anlysis of wavelets methods, in: Handbook in Numerical Analysis, Vol. VII, eds. P.G. Ciarlet and J.L. Lions (Elsevier, Amsterdam, 2000).
W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numer. 6 (1997) 55–228.
W. Dahmen and C.A. Micchelli, Using the refinement equation for evaluating integrals of wavelets, SIAM J. Numer. Anal. 30 (1993) 507–537.
J. Douglas, Jr. and T. Dupont, Galerkin method for parabolic equations, SIAM J. Numer. Anal. 7 (1970) 575–626.
G. Fix and G. Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory, Stud. Appl. Math. 48 (1969) 265–273.
J. Fröhlich and K. Schneider, An adaptive wavelet-vaguelette algorithm for the solution of PDEs, J. Comput. Phys. 130 (1997) 174–190.
R. Glowinski, W. Lawton, M. Ravachol and E. Tenenbaum, Wavelet solutions of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension, in: Computing Methods in Applied Science and Engineering, Paris, 1990, eds. R. Glowinski and A. Lichnewsky (SIAM, Philadelphia, PA, 1990), pp. 55–120.
S.M. Gomes and E. Cortina, Convergence estimates for the wavelet Galerkin method, SIAMJ. Numer. Anal. 33 (1996) 149–161.
T.N.T. Goodman and C.A. Micchelli, On refinement equations determined by Pólya frequency sequence, SIAM J. Math. Anal. 23 (1992) 766–784.
L. Gori and F. Pitolli, Multiresolution analysis based on certain compactly supported refinable functions, in: Approx. Opt., Cluj-Napoca, 1996, eds. G. Coman, W.W. Breckner and P. Blaga (Transilvania Press, 1997), pp. 81–90.
L. Gori and F. Pitolli, A class of totally positive refinable functions, Rend. Mat. Appl. (Ser. 7) 20 (2000) 305–322.
L. Gori and F. Pitolli, Refinable functions and positive operators, submitted.
M.L. Lo Cascio and F. Pitolli, Generalized cardinal interpolation by refinable functions: some numerical results, Note Mat. 15 (1995) 191–201.
A.K. Louis, P. Maass and A. Rieder, Wavelet. Theory and Applications, Series in Pure and Applied Mathematics (Wiley, Chichester, England, 1998).
Y. Maday, V. Perrier and J.C. Ravel, Adaptivité dinamique sur bases d'ondelettes pour l'approximation d'équations aux derivées partielles, C. R. Acad. Sci. Paris Sér. I 312 (1991) 405–410.
F. Pitolli, Refinement masks of Hurwitz type in the cardinal interpolation problem, Rend. Mat. Appl. (Ser. 7) 18 (1998) 549–563.
S. Qian and J. Weiss, Wavelets and numerical solution of partial differential equations, J. Comput. Phys. 106 (1993) 155–175.
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer-Verlag, Berlin, 1994).
I.J. Schoenberg, Cardinal Spline Interpolation, SIAM Monograph, Vol. 12 (SIAM, Philadelphia, PA, 1973).
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Gori, L., Pitolli, F. & Pezza, L. On the Galerkin Method Based on a Particular Class of Scaling Functions. Numerical Algorithms 28, 187–198 (2001). https://doi.org/10.1023/A:1014000418814
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DOI: https://doi.org/10.1023/A:1014000418814