Abstract
We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces S rd (Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the L p norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain.
Similar content being viewed by others
References
R. Adams, Sobolev Spaces (Academic Press, New York, 1975).
P. Alfeld, M. Neamtu and L.L. Schumaker, Dimension and local bases of homogeneous spline spaces, SIAM J. Math. Anal. 27 (1996) 1482–1501.
P. Alfeld and L.L. Schumaker, The dimension of bivariate spline spaces of smoothness r for degree d ⩾ 4r + 1, Constr. Approx. 3 (1987) 189–197.
C. de Boor, B-form basics, in: Geometric Modeling: Algorithms and New Trends, ed. G.E. Farin (SIAM, Philadelphia, PA, 1987) pp. 131–148.
C. de Boor and K. Höllig, Approximation power of smooth bivariate pp functions, Math. Z. 197 (1988) 343–363.
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994).
C.K. Chui, D. Hong and R.-Q. Jia, Stability of optimal order approximation by bivariate splines over arbitrary triangulations, Trans. Amer. Math. Soc. 347 (1995) 3301–3318.
C.K. Chui and M.J. Lai, On bivariate super vertex splines, Constr. Approx. 6 (1990) 399–419.
G. Farin, Triangular Bernstein–Bézier patches, Comput. Aided Geom. Design 3 (1986) 83–127.
D. Hong, Spaces of bivariate spline functions over triangulation, Approx. Theory Appl. 7 (1991) 56–75.
A. Ibrahim and L.L. Schumaker, Super spline spaces of smoothness r and degree d ⩾ 3r + 2, Constr. Approx. 7 (1991) 401–423.
R.Q. Jia, Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh, Trans. Amer. Math. Soc. 295 (1986) 199–212.
L.L. Schumaker, Spline Functions: Basic Theory (Wiley, New York, 1981).
L.L. Schumaker, On super splines and finite elements, SIAM J. Numer. Anal. 26 (1989) 997–1005.
E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, NJ, 1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lai, MJ., Schumaker, L.L. On the approximation power of bivariate splines. Advances in Computational Mathematics 9, 251–279 (1998). https://doi.org/10.1023/A:1018958011262
Issue Date:
DOI: https://doi.org/10.1023/A:1018958011262