Advertisement

BIT Numerical Mathematics

, Volume 54, Issue 4, pp 1099–1118 | Cite as

Cubic spline quasi-interpolants on Powell–Sabin partitions

  • A. Lamnii
  • M. Lamnii
  • H. Mraoui
Article

Abstract

By using the polarization identity, we propose a family of quasi-interpolants based on bivariate \({\fancyscript{C}}^1\) cubic super splines defined on triangulations with a Powell–Sabin refinement. Their spline coefficients only depend on a set of local function values. The quasi-interpolants reproduce cubic polynomials and have an optimal approximation order.

Keywords

Super spline Powell–Sabin splines Normalized B-splines Blossoms Polarization identity Quasi-interpolation 

Mathematics Subject Classification (2010)

41A15 65D05 65D17 

Notes

Acknowledgments

The authors thank the referees for the useful corrections and comments which improved the presentation of the paper.

References

  1. 1.
    Busch, J.R.: A note on Lagrange interpolation in \(\mathbb{R}^2\). Revista de la Unión Matemática Argentina 36, 33–38 (1990)MATHMathSciNetGoogle Scholar
  2. 2.
    de Casteljau, P.: Le Lissage. Hermes, Paris (1990)Google Scholar
  3. 3.
    Chena, S.-K., Liu, H.-W.: A bivariate \({\fancyscript{C}}^1\) cubic super spline space on Powell–Sabin triangulation. Comput. Math. Appl. 56, 1395–1401 (2008)Google Scholar
  4. 4.
    Chung, K.C., Yao, T.H.: On a lattices admitting unique Lagrange interpolation. SIAM J. Numer. Math. Anal. 14, 735–743 (1977)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Dierckx, P.: On calculating normalized Powell–Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Foucher, F., Sablonnière, P.: Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants on uniform meshes. Math. Comput. Simul. 77, 202–208 (2008)CrossRefMATHGoogle Scholar
  7. 7.
    Goldman, R.N.: Blossoming and knot insertion algorithms for B-spline curves. Comput. Aided Geom. Des. 7, 69–78 (1990)CrossRefMATHGoogle Scholar
  8. 8.
    Ibáñez, M.J., Lamnii, A., Mraoui, H., Sbibih, D.: Construction of spherical spline quasi-interpolants based on blossoming. J. Comput. Appl. Math. 234, 131–145 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Lai, M.-J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  10. 10.
    Lamnii, M., Mraoui, H., Tijini, A.: Construction of quintic Powell–Sabin spline quasi-interpolants based on blossoming. J. Comput. Appl. Math. 234, 190–209 (2013)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lamnii, M., Mraoui, H., Zidna, A.: Local quasi-interpolants based on special multivariate quadratic spline space over a refined quadrangulation. Appl. Math. Comput. 219, 10456–10467 (2013)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Lamnii, M., Mraoui, H., Tijini, A., Zidna, A.: A normalized basis for \({\fancyscript{C}}^1\) cubic super spline space on Powell–Sabin triangulation. Math. Comput. Simul. 99, 108–124 (2014)Google Scholar
  13. 13.
    Lamnii, M., Mazroui, A., Tijini, A.: Raising the approximation order of multivariate quasi-interpolants. BIT Numer. Math. (2014). doi: 10.1007/s10543-014-0470-8 MATHMathSciNetGoogle Scholar
  14. 14.
    Liu, H.-W., Chen, S.-K., Chen, Y.P.: Bivariate \({\fancyscript{C}}^1\) cubic spline space over Powell–Sabin type-1 refinement. J. Inf. Comput. Sci. 4, 151–160 (2007)Google Scholar
  15. 15.
    Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Des. 6, 323–358 (1989)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Sbibih, D., Serghini, A., Tijini, A.: Polar forms and quadratic spline quasi-interpolants on Powell Sabin partitions. Appl. Numer. Math. 59, 938–958 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Sbibih, D., Serghini, A., Tijini, A.: Normalized trivariate B-splines on Worsey-Piper split and quasi-interpolants. BIT Numer. Math. 52, 221–249 (2012)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Seidel, H.-P.: Symmetric recursive algorithms for surfaces: B-patches and the de Boor algorithm for polynomials over triangles. Constr. Approx. 7, 257–279 (1991)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Speleers, H.: A normalized basis for reduced Clough–Tocher splines. Comput. Aided Geom. Des. 27(9), 700–712 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Speleers, H.: A normalized basis for quintic Powell–Sabin splines. Comput. Aided Geom. Des. 27(6), 438–457 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Speleers, H.: Multivariate normalized Powell–Sabin B-splines and quasi-interpolants. Comput. Aided Geom. Des. 30, 2–19 (2013)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Speleers, H.: Construction of normalized B-splines for a family of smooth spline spaces over Powell–Sabin triangulations. Constr. Approx. 37(1), 41–72 (2013)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Speleers, H.: A family of smooth quasi-interpolants defined over Powell–Sabin triangulations. Technical Report TW617, KU Leuven (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Faculty of Science and Technology, University Hassan FirstSettatMorocco
  2. 2.Faculty of Science, University Mohammed FirstOujdaMorocco

Personalised recommendations