Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 

 

Bivariate $ C\sp 1$ quadratic finite elements and vertex splines


Authors: Charles K. Chui and Tian Xiao He
Journal: Math. Comp. 54 (1990), 169-187
MSC: Primary 65D07; Secondary 41A15, 41A63, 65N30
MathSciNet review: 993926
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Following work of Heindl and of Powell and Sabin, each triangle of an arbitrary (regular) triangulation $ \Delta $ of a polygonal region $ \Omega $ in $ {\mathbb{R}^2}$ is subdivided into twelve triangles, using the three medians, yielding the refinement $ \hat \Delta $ of $ \Delta $, so that $ {C^1}$ quadratic finite elements can be constructed. In this paper, we derive the Bézier nets of these elements in terms of the parameters that describe function and first partial derivative values at the vertices and values of the normal derivatives at the midpoints of the edges of $ \Delta $. Consequently, bivariate $ {C^1}$ quadratic (generalized) vertex splines on $ \Delta $ have an explicit formulation. Here, a generalized vertex spline is one which is a piecewise polynomial on the refined grid partition $ \hat \Delta $ and has support that contains at most one vertex of the original partition $ \Delta $ in its interior. The collection of all $ {C^1}$ quadratic generalized vertex splines on $ \Delta $ so constructed is shown to form a basis of $ S_2^1(\hat \Delta )$, the vector space of all functions on $ {C^1}(\Omega )$ whose restrictions to each triangular cell of the partition $ \hat \Delta $ are quadratic polynomials. A subspace with the basis given by appropriately chosen generalized vertex splines with exactly one vertex of $ \Delta $ in the interior of their supports, that reproduces all quadratic polynomials, is identified, and hence, has approximation order three. Quasi-interpolation formulas using this subspace are obtained. In addition, a constructive procedure that yields a locally supported basis of yet another subspace with dimension given by the number of vertices of $ \Delta $, that has approximation order three, is given.


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

  • [1] Peter Alfeld and L. L. Schumaker, The dimension of bivariate spline spaces of smoothness 𝑟 for degree 𝑑≥4𝑟+1, Constr. Approx. 3 (1987), no. 2, 189–197. MR 889554, 10.1007/BF01890563
  • [2] C. K. Chui and T. X. He, Bivariate $ {C^1}$ quadratic finite elements and vertex splines, CAT Report 147, Texas A&M University, 1987.
  • [3] C. K. Chui and M. J. Lai, On bivariate vertex splines, Multivariate approximation theory, III (Oberwolfach, 1985) Internat. Schriftenreihe Numer. Math., vol. 75, Birkhäuser, Basel, 1985, pp. 84–115. MR 890790
  • [4] Charles K. Chui and Ren Hong Wang, Multivariate spline spaces, J. Math. Anal. Appl. 94 (1983), no. 1, 197–221. MR 701458, 10.1016/0022-247X(83)90014-8
  • [5] G. Farin, Subsplines über Dreiecken, Dissertation, Braunschweig, 1979.
  • [6] Gerald Farin, Triangular Bernstein-Bézier patches, Comput. Aided Geom. Design 3 (1986), no. 2, 83–127. MR 867116, 10.1016/0167-8396(86)90016-6
  • [7] Gerhard Heindl, Interpolation and approximation by piecewise quadratic 𝐶¹-functions of two variables, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel-Boston, Mass., 1979, pp. 146–161. MR 560670
  • [8] Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985
  • [9] M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977), no. 4, 316–325. MR 0483304

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D07, 41A15, 41A63, 65N30

Retrieve articles in all journals with MSC: 65D07, 41A15, 41A63, 65N30


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-0993926-3
Keywords: Bivariate splines, interpolation, quasi-interpolation, macroelements, vertex splines
Article copyright: © Copyright 1990 American Mathematical Society