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BIT Numerical Mathematics

Erschienen in:

01.09.2015

Superconvergent \(C^1\) cubic spline quasi-interpolants on Powell-Sabin partitions

verfasst von: Driss Sbibih, Abdelhafid Serghini, Ahmed Tijini, Ahmed Zidna

Erschienen in: BIT Numerical Mathematics | Ausgabe 3/2015

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Abstract

In this paper we introduce a B-spline representation of the cubic Hermite Powell-Sabin interpolant of any polynomial or any piecewise polynomial, over Powell-Sabin partitions of class at least \(C^1\), in terms of their polar forms. We use this B-spline representation for constructing several superconvergent discrete cubic spline quasi-interpolants which approximate a function \(f\) better than the superconvergent quadratic ones developed in one of our recent published papers. The new results presented in this work are an improvement and a generalization of those studied recently in the literature. We also illustrate by numerical examples that global errors and cubature rules based on these cubic Powell-Sabin spline quasi-interpolants are positively affected by the superconvergence phenomenon.

Vorheriger Artikel Between moving least-squares and moving least-

Nächster Artikel Convergence properties of a quadrature formula of Clenshaw–Curtis type for the Gegenbauer weight function

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de Boor, C.: The quasi-interpolant as a tool in elementary polynomial spline theory. In: Berens, H., Cheney, E.W., Lorentz, G.G., Schumaker, L.L. (eds.) Approximation Theory I, pp. 269–276. Academic Press, New York (1973)

Carnicer, J.M., Gasca, M.: Planar configurations with simple Lagrange formulae. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in CAGD: Oslo 2000, pp. 55–62. Vanderbilt University Press, Nashville (2001)

Chen, S.-K., Liu, H.-W.: A bivariate \(C^1\) cubic super spline on Powell-Sabin triangulation. Comput. Math. Appl. 56, 1395–1401 (2008)MATHMathSciNetCrossRef

Chung, K.C., Yao, T.H.: On lattices admitting unique Lagrange interpolation. SIAM J. Numer. Math. Anal. 14, 735–743 (1977)MATHMathSciNetCrossRef

Dierckx, P.: On calculating normalized Powell-Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)MATHMathSciNetCrossRef

Fangy, Q., Yamamoto, T.: Superconvergence of finite difference approximations for convection diffusion problems. Numer. Linear Algebra Appl. 8, 99–110 (2001)MathSciNetCrossRef

Foucher, F., Sablonnière, P.: Quadratic spline quasi-interpolants and collocation methods. Math. Comp. Simul. 79, 3455–3465 (2009)MATHCrossRef

Lai, M.J., Wenston, P.: Bivariate splines for fluid flows. Comput. Fluids 33, 1047–1073 (2004)MATHMathSciNetCrossRef

Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press, New York (2007)MATHCrossRef

10.

Lamnii, M., Mraoui, H., Tijini, A., Zidna, A.: A normalized basis for \(C^1\) cubic super spline space on Powell-Sabin triangulation. Math. Comput. Simul. 99, 108124 (2014)MathSciNetCrossRef

11.

Liang, X.-Z.: On the interpolation and approximation in several variables, Postgraduate Thesis, Jilin University, (1965)

12.

Liu, H.-W., Chen, S.-K., Chen, Y.P.: Bivariate \(C^1\) cubic spline space over Powell-Sabin type-1 refinement. J. Inform. Comput. Sci. 4, 151–160 (2007)

13.

Liu, J., Zhang, X., Yu, D.: Superconvergence and ultraconvergence of Newton-Cotes rules for supersingular integrals. J. Comput. Appl. Math. 233, 2841–2854 (2010)MathSciNetCrossRef

14.

Liu, D., Wu, J., Yu, D.: The superconvergence of the Newton-Cotes rule for Cauchy principal value integrals. J. Comput. Appl. Math. 235, 696–707 (2010)MATHMathSciNetCrossRef

15.

Maes, J.: Powell-Sabin spline based multilevel preconditioners for elliptic partial differential equations. Phd thesis, Katholieke Universiteit Leuven, Belgium, (2006)

16.

Maes, J., Vanraes, E., Dierckx, P., Bultheel, A.: On the stability of normalized Powell-Sabin B-splines. J. Comput. Appl. Math. 170(1), 181196 (2004)MathSciNetCrossRef

17.

Remogna, S.: Constructing good coefficient functionals for bivariate \(C^1\) quadratic spline quasi-interpolants. In: Daehlen, M., et al. (eds.) Mathematical Methods for Curves and Surfaces, LNCS 5862, pp. 329–346. Springer, Berlin (2010)CrossRef

18.

Remogna, S.: Bivariate \(C^2\) cubic spline quasi-interpolants on uniform Powell-Sabin triangulations of a rectangular domain. Adv. Comput. Math. 36, 39–65 (2012)MATHMathSciNetCrossRef

19.

Sablonnière, P., Sbibih, D., Tahrichi, M.: Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions, BIT Numer. Math. 53 (1), 175–192, 743–750 (2013)

20.

Sbibih, D., Serghini, A., Tijini, A.: Superconvergent quadratic splines quasi-interpolants over Powell-Sabin triangulation, Appl. Numer. Math. (submitted 2013)

21.

Sbibih, D., Serghini, A., Tijini, A.: Superconvergent trivariate quadratic spline quasi-interpolants on Worsey-Piper split. J. Comput. Appl. Math. (submitted 2013)

22.

Sbibih, D., Serghini, A., Tijini, A.: Polar forms and quadratic splines quasi-interpolants over Powell-Sabin triangulation. Appl. Numer. Math. 59, 938–958 (2009)MATHMathSciNetCrossRef

23.

Sbibih, D., Serghini, A., Tijini, A.: Normalized trivariate B-splines on Worsey-Piper split and quasi-interpolants. BIT Numer. Math. 52, 221–249 (2012)MATHMathSciNetCrossRef

24.

Sloan, J.H., Wendland, W.L.: Superconvergence of the Galerkin iterates for integral equations of the second kind. J. Integral Equ. 9, 1–23 (1985)MATH

25.

Speleers, H.: A normalized basis for quintic Powell-Sabin splines. Comput. Aided Geom. Des. 27, 438–457 (2010)MATHMathSciNetCrossRef

26.

Speleers, H.: A normalized basis for reduced Clough Tocher splines. Comput. Aided Geom. Des. 27, 700–712 (2010)MATHMathSciNetCrossRef

27.

Speleers, H.: Multivariate normalized Powell-Sabin B-splines and quasi-interpolants. Comput. Aided Geom. Des. 30, 2–19 (2013)MATHMathSciNetCrossRef

28.

Speleers, H.: Construction of normalized B-splines for a family of smooth spline spaces over Powell-Sabin triangulations. Constr. Approx. 37, 41–72 (2013)MATHMathSciNetCrossRef

29.

Wahlbin, L.B.: Local behaviour in finite element methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, Vol II, Finite Element Methods (Part 1), pp. 355–522. Elsevier, Amsterdam (1991)

30.

Zhang, X., Wu, J., Yu, D.: Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications. J. Comput. Appl. Math. 223, 598–613 (2009)MATHMathSciNetCrossRef

Titel: Superconvergent cubic spline quasi-interpolants on Powell-Sabin partitions
verfasst von: Driss Sbibih
Abdelhafid Serghini
Ahmed Tijini
Ahmed Zidna
Publikationsdatum: 01.09.2015
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 3/2015
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-014-0523-z

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