zbMATH — the first resource for mathematics

\(C^ r\)-finite elements of Powell-Sabin type on the three direction mesh. (English) Zbl 0867.65002
The triangulation \(\tau\) generated by a uniform three direction mesh of the plane and the Powell-Sabin subtriangulation \(\tau_6\) obtained by subdividing each triangle \(T\in\tau\) by connecting each vertex to the midpoint of the opposite side are considered. The existence of Hermite interpolation schemes in a subspace of \(S^r_n (\tau_6)\) for lower degrees (i.e. \(n=2r+1\) for \(r\) even and \(n=2r\) for \(r\) odd) is proven. Some results on the Bernstein-Bézier form of polynomials on triangles which is used for representing splines on the triangulation \(\tau_6\) are given. The construction of Powell-Sabin finite elements and the solution of the Hermite interpolation problem of order \(r\) are discussed. The interpolation error is estimated.
Reviewer: V.Burjan (Praha)

65D05 Numerical interpolation
65D07 Numerical computation using splines
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
41A05 Interpolation in approximation theory
41A15 Spline approximation
Full Text: DOI
[1] R. Arcangeli and J. L. Gout, Sur l’évaluation de l’erreur d’interpolation de Lagrange dans un ouvert de \(\mathbb{R}\) n , RAIRO Anal. Num. 10 (1976) 5–27. · Zbl 0337.65008
[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.
[3] C. de Boor and K. Höllig, Approximation power of smooth bivariate pp function, Math. Z. 197 (1988) 343–363. · Zbl 0616.41010 · doi:10.1007/BF01418335
[4] C. de Boor, K. Höllig and S. Riemenschneider,Box Splines (Springer, New York, 1993).
[5] C. K. Chui,Multivariate Splines (SIAM, Philadelphia, PA, 1988).
[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.
[7] C. K. Chui and M. J. Lai, On bivariate supervertex splines, Constr. Approx. 6 (1990) 399–419. · Zbl 0726.41012 · doi:10.1007/BF01888272
[8] P. G. Ciarlet,The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1977).
[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. · Zbl 0875.65086
[10] G. Farin, Triangular Bernstein-Bézier patches, Computer Aided Geometric Design 2 (1986) 83–127. · doi:10.1016/0167-8396(86)90016-6
[11] J. L. Gout, Estimation de l’erreur d’interpolation d’Hermite dans \(\mathbb{R}\) n , Numer. Math. 28 (1977) 407–429. · Zbl 0365.65007 · doi:10.1007/BF01404344
[12] A. K. Ibrahim and L. L. Schumaker, Super splines spaces of smoothnessr and degreedr+2, Constr. Approx. 7 (1991) 401–423. · Zbl 0739.41011 · doi:10.1007/BF01888166
[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.
[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.
[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. · Zbl 0736.41003
[16] M. Laghchim-Lahlou, Eléments finis composites de classeC k dans \(\mathbb{R}\)2, Thèse de Doctorat, INSA de Rennes (1991). · Zbl 0736.41003
[17] M. Laghchim-Lahlou and P. Sablonnière, Triangular finite elements of HCT type and classC \(\rho\) , Adv. Comput. Math. 2 (1994) 101–122. · Zbl 0832.65003 · doi:10.1007/BF02519038
[18] M. Laghchim-Lahlou and P. Sablonnière, Quadrilateral finite elements of FVS type and classC \(\rho\) , Numer. Math. 70 (1995) 229–243. · Zbl 0824.41012 · doi:10.1007/s002110050117
[19] A. Le Méhauté, Construction of surfaces of classC k on a domain \(\Omega\) \(\mathbb{R}\)2 after triangulation, in:Multivariate Approximation Theory II, eds. W. Schempp and K. Zeller, Internat. Series Numer. Math. 61 (Birkhäuser, Basel, 1982) pp. 223–240.
[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.
[21] M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximation on triangles, ACM Trans. Math. Software 3(4) (1977) 316–325. · Zbl 0375.41010 · doi:10.1145/355759.355761
[22] P. Sablonnière, Composite finite elements of classC k , J. Comput. Appl. Math. 12/13 (1985) 541–550. · Zbl 0587.41004 · doi:10.1016/0377-0427(85)90047-0
[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.
[25] P. Sablonnière, Error bounds for Hermite interpolation by quadratic splines, IMA J. Numer. Anal. 7 (1987) 495–508. · Zbl 0633.41004 · doi:10.1093/imanum/7.4.495
[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. · Zbl 0772.65006
[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.
[28] L. L. Schumaker, On super splines and finite elements, SIAM J. Numer. Anal. 26 (1989) 997–1005. · Zbl 0679.41008 · doi:10.1137/0726055
[29] A. Ženišek, A general theorem on triangular finiteC m -elements, RAIRO Anal. Numer. 2 (1974) 119–127.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.