Mathematics of Computation

Macro-elements and stable local bases for splines on Powell-Sabin triangulations

Author(s): Ming-Jun Lai; Larry L. Schumaker.
Journal: Math. Comp. 72 (2003), 335-354.
MSC (2000): Primary 41A15, 65M60, 65N30
Posted: July 22, 2001
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Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements are useful for solving boundary-value problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Powell-Sabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power.

1.: Alfeld, P., Schumaker, L. L., The dimension of bivariate spline spaces of smoothness for degree $d \ge 4r+1$ , Constr. Approx., 3 (1987), 189-197. MR 88e:41025
2.: Chui, C. K., Hong, D., Jia, R. Q., Stability of optimal-order approximation by bivariate splines over arbitrary triangulations, Trans. Amer. Math. Soc., 347(1995), 3301-3318. MR 96d:41012
3.: Davydov, O., Schumaker, L. L., On stable local bases for bivariate polynomial splines, Constr. Approx., to appear.
4.: Farin, G., Curves and Surfaces for Computer Aided Geometric Design, Academic Press (NY), 1988. MR 90c:65014
5.: Hoschek, J., Lasser, D., Fundamentals of Computer Aided Geometric Design, A. K. Peters (Boston MA), 1993. MR 94i:65003
6.: Ibrahim, A., Schumaker, L. L., Super spline spaces of smoothness and degree $d\ge 3r+2$ , Constr. Approx., 7(1991), 401-423. MR 92k:41017
7.: Jia, R. Q., Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh, Trans. Amer. Math. Soc., 295(1986), 199-212. MR 88d:41018
8.: Laghchim-Lahlou, M., Eléments finis composites de classe $C^{k}$ dans $\mathbb{R}^{2}$ , Thèse de Doctorat, INSA de Rennes, 1991.
9.: Laghchim-Lahlou, M., Composite $C^{r}$ -triangular finite elements of PS type on a three direction mesh, (Curves and Surfaces), P.-J. Laurent, A. Le Méhauté, and L. L. Schumaker (eds.), Vanderbilt Univ. Press (Nashville), 1991, 275-278. CMP 91:17
10.: Laghchim-Lahlou, M., $C^{r}$ -finite elements of Powell-Sabin type on the three direction mesh, Advances in Comp. Math., 6(1996), 191-206. MR 97j:65031
11.: Laghchim-Lahlou, M., The $C^{r}$ -fundamental splines of Clough-Tocher and Powell-Sabin types for Lagrange interpolation on a three-direction mesh, Advances in Comp. Math., 8(1998), 353-366. MR 99j:65015
12.: Laghchim-Lahlou, M., Interpolation d'Hermite ou de Lagrange dans des espaces de supersplines composites dan le plan, Thesis, Univ. Cadi Ayyad, Marrakesh, 1998.
13.: Laghchim-Lahlou, M., Sablonnière, P., Triangular finite elements of HCT type and class $C^{\rho }$ , Advances in Comp. Math., 2(1994), 101-122. MR 95d:65013
14.: Laghchim-Lahlou, M., Sablonnière, P., Quadrilateral finite elements of FVS type and class $C^{\rho }$ , Numer. Math., 70(1995), 229-243. MR 96e:65006
15.: Lai, M.-J., On $C^{2}$ quintic spline functions over triangulations of Powell-Sabin's type, J. Comput. Appl. Math., 73(1996), 135-155. MR 98a:41002
16.: Lai, M.-J., Schumaker, L. L., On the approximation power of bivariate splines, Advances in Comp. Math., 9(1998), 251-279. MR 2000b:41010
17.: Lai, M.-J., Schumaker, L. L., On the approximation power of splines on triangulated quadrangulations, SIAM J. Numer. Anal., 36(1999), 143-159. MR 99h:41014
18.: Lai, M.-J., Schumaker, L. L., Macro-elements and stable local bases for spaces of splines on Clough-Tocher triangulations, Numerische Math., to appear.
19.: Powell, M. J. D., Sabin, M. A., Piecewise quadratic approximations on triangles, ACM Trans. Math. Software, 3(1977), 316-325. MR 58:3319
20.: Sablonnière, P., Composite finite elements of class $C^{k}$ , J. Comp. Appl. Math., 12(1985), 541-550. CMP 17:14
21.: Sablonnière, P., Eléments finis triangulaires de degré 5 et de classe $C^{2}$ , (Computers and Computing), P. Chenin et al (eds.), Wiley (New York), 1986, 111-115.
22.: Sablonnière, P., Composite finite elements of class $C^{2}$ , Topics in Multivariate Approximation, C. K. Chui, L. L. Schumaker, and F. Utreras (eds.), Academic Press (New York), 1987, 207-217. MR 90b:65212
23.: Sablonnière, P., Error bounds for Hermite interpolation by quadratic splines on an $\alpha$ -triangulation, IMA J. Numer. Anal., 7(1987), 495-508. MR 90a:65029
24.: Sablonnière, P., Laghchim-Lahlou, M., Eléments finis polynomiaux composés de classe $C\sp{r}$ , C. R. Acad. Sci. Paris Sr. I Math., 316(1993), 503-508. MR 94a:65059

Retrieve articles in Mathematics of Computation with MSC (2000): 41A15, 65M60, 65N30

Ming-Jun Lai
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: mjlai@math.uga.edu

Larry L. Schumaker
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: s@mars.cas.vanderbilt.edu

DOI: 10.1090/S0025-5718-01-01379-5
PII: S 0025-5718(01)01379-5
Keywords: Macro-elements, stable bases, spline spaces
Received by editor(s): March 8, 2000
Received by editor(s) in revised form: January 31, 2001
Posted: July 22, 2001
Additional Notes: The first author was supported by the National Science Foundation under grant DMS-9870187
The second author was supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160
Copyright of article: Copyright 2001, American Mathematical Society

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