Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The structure of multivariate superspline spaces of high degree


Authors: Peter Alfeld and Maritza Sirvent
Journal: Math. Comp. 57 (1991), 299-308
MSC: Primary 65D07; Secondary 41A15
DOI: https://doi.org/10.1090/S0025-5718-1991-1079007-2
MathSciNet review: 1079007
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider splines (of global smoothness r, polynomial degree d, in a general number k of independent variables, defined on a k-dimensional triangulation $ \mathcal{T}$ of a suitable domain $ \Omega $) which are $ r{2^{k - m - 1}}$-times differentiable across every m-face $ (m = 0, \cdots ,k - 1)$ of a simplex in $ \mathcal{T}$. For the case $ d > r{2^k}$ we identify a structure that allows the construction of a minimally supported basis.


References [Enhancements On Off] (What's this?)

  • [1] P. Alfeld, Scattered data interpolation in three or more variables, Mathematical Methods in Computer Aided Geometric Design (Tom Lyche and Larry L. Schumaker eds.), Academic Press, New York, 1989, pp. 1-34. MR 1022695 (90j:65015)
  • [2] -, A bivariate $ {C^2}$ Clough-Tocher scheme, Comput. Aided Geom. Design 1 (1984), 257-267.
  • [3] Peter Alfeld, B. Piper, and L. L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree $ d \geq 4r + 1$, Comput. Aided Geom. Design 4 (1987), 105-123. MR 898027 (88d:65019)
  • [4] Peter Alfeld and L. L. Schumaker, The dimension of bivariate spline spaces of smoothness r for degree $ d \geq 4r + 1$, Constr. Approx. 3 (1987), 189-197. MR 889554 (88e:41025)
  • [5] Peter Alfeld and M. Sirvent, A recursion formula for the dimension of superspline spaces of smoothness r and degree $ d > r{2^k}$, Approximation Theory V (W. Schempp and K. Zeller, eds.), (Proceedings of the Oberwolfach Meeting, February 12-18, 1989), Birkhäuser Verlag, Basel, 1989, pp. 1-8. MR 1034290 (91k:41018)
  • [6] C. de Boor, B-form basics, Geometric Modeling: Algorithms and New Trends (G. E. Farin, ed.), SIAM, Philadelphia, PA, 1987, pp. 131-148. MR 936450
  • [7] C. K. Chui, Multivariate splines, SIAM, Philadelphia, PA, 1988. MR 1033490 (92e:41009)
  • [8] C. K. Chui and M. J. Lai, On multivariate vertex splines and applications, Topics in Multivariate Approximation (C. K. Chui, L. L. Schumaker, and F. Utreras, eds.), Academic Press, New York, 1987, pp. 19-36. MR 924820 (89h:41024)
  • [9] -, On bivariate vertex splines, Multivariate Approximation Theory III (W. Schempp and K. Zeller, eds.), Birkhäuser Verlag, Basel, 1985, pp. 84-115. MR 890790 (88m:41002)
  • [10] G. Farin, Triangular Bernstein-Bézier patches, Comput. Aided Geom. Design 3 (1986), 83-128. MR 867116 (87k:65014)
  • [11] A. Ibrahim and L. L. Schumaker, Superspline spaces of smoothness r and degree $ d \geq 3r + 2$, Constr. Approx., 1991. MR 1120412 (92k:41017)
  • [12] C. L. Lawson, Properties of n-dimensional triangulations, Comput. Aided Geom. Design 3 (1986), 231-247. MR 904930 (88h:68084)
  • [13] J. Morgan and R. Scott, A nodal basis for $ {C^1}$ piecewise polynomials of degree $ n \geq 5$, Math. Comp. 29 (1975), 736-740. MR 0375740 (51:11930)
  • [14] L. L. Schumaker, On the dimension of spaces of piecewise polynomials in two variables, Multivariate Approximation Theory (W. Schempp and K. Zeller, eds.), Birkhäuser Verlag, Basel, 1979, pp. 396-412. MR 560683 (81d:41011)
  • [15] -, Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky Mountain J. Math. 14 (1984), 251-264. MR 736177 (85h:41091)
  • [16] -, On super splines and finite elements, SIAM J. Numer. Anal. 26 (1989), 997-1005. MR 1005521 (90g:65016)
  • [17] G. Strang, Piecewise polynomials and the finite element method, Bull. Amer. Math. Soc. 79 (1973), 1128-1137. MR 0327060 (48:5402)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D07, 41A15

Retrieve articles in all journals with MSC: 65D07, 41A15


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1079007-2
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society