Summary
We consider the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally. To date, dimension formulae for such spaces have been established only whend≧3r+2, (except for the special case wherer=1 andd=4). In this paper we establish dimension formulae for allr≧1 andd=3r+1 for almost all triangulations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alfeld, P.: On the dimension of multivariate piecewise polynomials. In: Griffiths, D.F., Watson, G.A. (eds.) Numerical analysis, pp. 1–23. Essex: Longman Scientific and Technical, 1985
Alfeld, P.: A case study of multivariate piecewise polynomials. In: Farin, G. (ed.) Geometric modelling, pp. 149–160. SIAM, Philadelphia, 1987
Alfeld, P., Piper, B., Schumaker, L.L.: An explicit basis forC 1 quartic bivariate splines. SIAM J. Numer. Anal.24, 891–911 (1987)
Alfeld, P., Piper, B., Schumaker, L.L.: Minimally supported bases for spaces of bivariate piecewise polynomials of smoothnessr and degreed≧4r+1. Comput. Aided Geom. Des.4, 105–123 (1987)
Alfeld, P., Piper, B., Schumaker, L.L.: Spaces of bivariate splines on triangulations with holes. J. Approximation Theory Appl.3, 1–10 (1987)
Alfeld, P., Schumaker, L.L.: The dimension of bivariate spline spaces of smoothnessr for degreed≧4r+1. Constr. Approximation3, 189–197 (1987)
Billera, L.J.: Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. Am. Math. Soc.310, 325–340 (1988)
Chui, C.K., He, T.X.: On the dimension of bivariate super spline spaces. Math. Comput. (to appear)
Chui, C.K., Lai, M.J.: On bivariate vertex splines. In: Schempp, W., Zeller, K. (eds.) Multivariate approximation theory, III, pp. 84–115. Basel: Birkhäuser 1985
Chui, C.K., Lai, M.J.: Multivariate vertex splines and finite elements. J. Approximation Theory (to appear)
Hong, Dong: Spaces of multivariate spline functions over triangulations. Master's Thesis, Zheziang Univ., 1988
Ibrahim, A.Kh., Schumaker, L.L.: Superspline spaces of smoothnessr and degreed≧3r+2. Constr. Approximation (to appear)
Schumaker, L.L.: On the dimension of spaces of piecewise polynomials in two variables. In: Schempp, W., Zeller, K. (eds.) Multivariate approximation theory, pp. 396–412. Basel: Birkhäuser 1979
Schumaker, L.L.: Bounds on the dimension of spaces of multivariate piecewise polynomials. Rocky Mt. J. Math.14, 251–264 (1984)
Schumaker, L.L.: On spaces of piecewise polynomials in two variables. In: Singh, S.P., Burry, J.H.W., Watson, B. (eds.) Approximation theory and spline functions, pp. 151–197. Dordrecht: Reidel 1984
Schumaker, L.L.: Dual bases for spline spaces on cells. Comput. Aided Geom. Des.5, 277–284 (1988)
Schumaker, L.L.: On super splines and finite elements. SIAM J. Numer. Anal.26, 997–1005 (1989)
Author information
Authors and Affiliations
Additional information
Dedicated to R. S. Varga on the occasion of his sixtieth birthday
Supported in part by National Science Foundation Grant DMS-8701121
Supported in part by National Science Foundation Grant DMS-8602337
Rights and permissions
About this article
Cite this article
Alfeld, P., Schumaker, L.L. On the dimension of bivariate spline spaces of smoothnessr and degreed=3r+1. Numer. Math. 57, 651–661 (1990). https://doi.org/10.1007/BF01386434
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01386434