Abstract
We consider spaces of piecewise polynomials of degreed defined over a triangulation of a polygonal domain and possessingr continuous derivatives globally. Morgan and Scott constructed a basis in the case wherer=1 andd≥5. The purpose of this paper is to extend the dimension part of their result tor≥0 andd≥4r+l. We use Bézier nets as a crucial tool in deriving the dimension of such spaces.
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References
P. Alfeld (1985):On the dimension of multivariate piecewise polynomial functions. Proceedings of the Biennial Dundee Conference on Numerical Analysis, June 25–28. London: Pitman.
P. Alfeld.A case study of multivariate piecewise polynomials. In: Geometric Modelling (G. Farin, ed.). Philadelphia: SIAM. To appear.
P. Alfeld, B. Piper, L. L. Schumaker (submitted).Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degrered≥4r+1.
W. Böhm, G. Farin, J. Kahmann (1984):A survey of curve and surface methods in CAGD. Comput. Aided Geom. Design,1: 1–60.
C. de Boor (1986):B-form basics In: Geometric Modelling (G. Farin, ed.). Philadelphia: SIAM. To appear.
J. Morgan, R. Scott (1975):A nodal basis for C 1 piecewise polynomials of degree n≥5. Math. Comp.,29: 736–740.
J. Morgan, R. Scott (1975):The dimension of the space of C 1 piecewise polynomials. Unpublished manuscript.
L. L. Schumaker (1979):Lower bounds for the dimension of spaces of piecewise polynomials in two variables. In: Multivariate Approximation Theory (W. Schempp, K. Zeller, eds.). Birkhäuser-Verlag, pp. 396–412.
L. L. Schumaker (1984):On spaces of piecewise polynomials in two variables. In: Splines and Approximation Theory (Singhet al., eds.). Dordrecht: Reidel, pp. 151–197.
L. L. Schumaker (1984):Bounds on the dimension of spaces of multivariate piecewise polynomials. Rocky Mountain J. Math.,14: 251–264.
G. Strang, Fix, G. J. (1973): An Analysis of the Finite Element Method. Englewood Cliffs, NJ: Prentice-Hall.
A. Zeníšek (1970):Interpolation polynomials on the triangle. Numer. Math.,15: 283–296.
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Communicated by Klaus Höllig.
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Alfeld, P., Schumaker, L.L. The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1. Constr. Approx 3, 189–197 (1987). https://doi.org/10.1007/BF01890563
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DOI: https://doi.org/10.1007/BF01890563