Skip to main content
SpringerLink
Log in

C r-finite elements of Powell-Sabin type on the three direction mesh

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Letτ be the triangulation generated by a uniform three direction mesh of the plane. Letτ 6 be the Powell-Sabin subtriangulation obtained by subdividing each triangleT ∈τ by connecting each vertex to the midpoint of the opposite side.

Given a smooth functionu, we construct a piecewise polynomial functionυ ∈C r (ℝ2) of degreen=2r (resp. 2r+1) forr odd (resp. even) in each triangle ofτ 6, interpolating derivatives ofu up to orderr at the vertices ofτ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Arcangeli and J. L. Gout, Sur l'évaluation de l'erreur d'interpolation de Lagrange dans un ouvert de ℝn, RAIRO Anal. Num. 10 (1976) 5–27.

    Google Scholar 

  2. C. de Boor, B-net basics, in:Geometric Modelling, Algorithms and New Trends, ed. G. E. Farin (SIAM, Philadelphia, PA, 1987) pp. 131–148.

    Google Scholar 

  3. C. de Boor and K. Höllig, Approximation power of smooth bivariate pp function, Math. Z. 197 (1988) 343–363.

    Article  Google Scholar 

  4. C. de Boor, K. Höllig and S. Riemenschneider,Box Splines (Springer, New York, 1993).

    Google Scholar 

  5. C. K. Chui,Multivariate Splines (SIAM, Philadelphia, PA, 1988).

    Google Scholar 

  6. C. K. Chui and M. J. Lai, On multivariate vertex splines and applications, in:Topics in Multivariate Approximation, eds. C. K. Chui, L. L. Schumaker and F. Utreras (Academic Press, New York, 1987) pp. 19–36.

    Google Scholar 

  7. C. K. Chui and M. J. Lai, On bivariate supervertex splines, Constr. Approx. 6 (1990) 399–419.

    Article  Google Scholar 

  8. P. G. Ciarlet,The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1977).

    Google Scholar 

  9. P. G. Ciarlet, Basic error estimates for elliptic problems, in:Handbook of Numerical Analysis, Vol. II,Finite Element Method (Part 1), eds. P. G. Ciarlet and J. L. Lions (North-Holland, 1991) pp. 19–351.

  10. G. Farin, Triangular Bernstein-Bézier patches, Computer Aided Geometric Design 2 (1986) 83–127.

    Article  Google Scholar 

  11. J. L. Gout, Estimation de l'erreur d'interpolation d'Hermite dans ℝn, Numer. Math. 28 (1977) 407–429.

    Article  Google Scholar 

  12. A. K. Ibrahim and L. L. Schumaker, Super splines spaces of smoothnessr and degreed⩾3r+2, Constr. Approx. 7 (1991) 401–423.

    Article  Google Scholar 

  13. M. Laghchim-Lahlou and P. Sablonnière, Composite quadrilateral finite elements of classC r, in:Mathematical Methods in CAGD, eds. T. Lyche and L. L. Schumaker (Academic Press, New York, 1989) pp. 413–418.

    Google Scholar 

  14. M. Laghchim-Lahlou and P. Sablonnière,C r finite elements of HCT, PS and FVS types, in:Proc. 5th Int. Symp. on Numerical Methods in Engineering, Vol. 2, eds. R. Gruber, J. Periau and R. P. Shaw (Springer, Berlin, 1989) pp. 163–168.

    Google Scholar 

  15. M. Laghchim-Lahlou, CompositeC r-triangular finite elements of PS type on a three direction mesh, in:Curves and Surfaces, eds. P. J. Laurent, A. Le Méhauté and L. L. Schumaker (Academic Press, 1991) pp. 275–278.

  16. M. Laghchim-Lahlou, Eléments finis composites de classeC k dans ℝ2, Thèse de Doctorat, INSA de Rennes (1991).

  17. M. Laghchim-Lahlou and P. Sablonnière, Triangular finite elements of HCT type and classC ρ, Adv. Comput. Math. 2 (1994) 101–122.

    Google Scholar 

  18. M. Laghchim-Lahlou and P. Sablonnière, Quadrilateral finite elements of FVS type and classC ρ, Numer. Math. 70 (1995) 229–243.

    Article  Google Scholar 

  19. A. Le Méhauté, Construction of surfaces of classC k on a domain Ω ⊂ ℝ2 after triangulation, in:Multivariate Approximation Theory II, eds. W. Schempp and K. Zeller, Internat. Series Numer. Math. 61 (Birkhäuser, Basel, 1982) pp. 223–240.

    Google Scholar 

  20. A. Le Méhauté, A finite element approach to surface reconstruction, in:Computation of Curves and Surfaces, eds. W. Dahmen, M. Gasca and C. A. Micchelli, NATO ASI Series 307 (Kluwer, Dordrecht, 1990) pp. 237–274.

    Google Scholar 

  21. M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximation on triangles, ACM Trans. Math. Software 3(4) (1977) 316–325.

    Article  Google Scholar 

  22. P. Sablonnière, Composite finite elements of classC k, J. Comput. Appl. Math. 12/13 (1985) 541–550.

    Article  Google Scholar 

  23. P. Sablonnière, Eléments finis triangulaires de degré 5 et de classeC 2, in:Computers and Computing, eds. P. Chenin, C. di Crescenzo and F. Robert (Wiley-Masson, 1986) pp. 111–115.

  24. P. Sablonnière, Composite finite elements of classC 2, in:Topics in Multivariate Approximation Theory, eds. C. K. Chui, L. L. Schumaker and F. Utreras (Academic Press, New York, 1987) pp. 207–217.

    Google Scholar 

  25. P. Sablonnière, Error bounds for Hermite interpolation by quadratic splines, IMA J. Numer. Anal. 7 (1987) 495–508.

    Google Scholar 

  26. P. Sablonnière and M. Laghchim-Lahlou, Eléments finis polynômiaux composés de classeC r, C. R. Acad. Sci. Paris, t. 316, Série I (1993) 503–508.

    Google Scholar 

  27. L. L. Schumaker, On the dimension of spaces of piecewise polynomials in two variables, in:Multivariate Approximation Theory, eds. W. Schempp and K. Zeller, Internat. Series Numer. Math. 51 (Birkhäuser, Basel, 1979) pp. 396–412.

    Google Scholar 

  28. L. L. Schumaker, On super splines and finite elements, SIAM J. Numer. Anal. 26 (1989) 997–1005.

    Article  Google Scholar 

  29. A. Ženišek, A general theorem on triangular finiteC m-elements, RAIRO Anal. Numer. 2 (1974) 119–127.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Faculté des sciences-Semlalia, Université Cadi Ayyad, BP S 15, 40000, Marrakech, Morocco

    M. Laghchim-Lahlou

  2. Laboratoire LANS, INSA, 20 avenue des Buttes de Coësmes, 35043, Rennes Cédex, France

    P. Sablonnière

Authors
  1. M. Laghchim-Lahlou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Sablonnière
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by L.L. Schumaker

Rights and permissions

Reprints and permissions

About this article

Cite this article

Laghchim-Lahlou, M., Sablonnière, P. C r-finite elements of Powell-Sabin type on the three direction mesh. Adv Comput Math 6, 191–206 (1996). https://doi.org/10.1007/BF02127703

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02127703

Keywords

AMS subject classification

Search

Navigation