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On the dimension of bivariate superspline spaces

Authors: Charles K. Chui and Tian Xiao He
Journal: Math. Comp. 53 (1989), 219-234
MSC: Primary 41A15
Corrigendum: Math. Comp. 55 (1990), 407-409.
Corrigendum: Math. Comp. 55 (1990), 407-409.
MathSciNet review: 969483
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Abstract: A bivariate piecewise polynomial function of total degree d on some grid partition $ \Delta $ that has rth order continuous partial derivatives everywhere may have higher-order partial derivatives at the vertices of the grid partition. In finite element considerations and in the construction of vertex splines, it happens that only those functions with continuous partial derivatives of order higher than r at the vertices are needed to give the same full approximation order as the entire space of piecewise polynomials. This is certainly the case for $ d \geq 4r + 1$. Such piecewise polynomial functions are called supersplines. This paper is devoted to the study of the dimension of certain superspline spaces. Since an exact dimension would have to depend on the geometric structure of the partition $ \Delta $, we will give only upper and lower bounds. We will show, however, that the lower bound value is sharp for all quasi-crosscut partitions; and under suitable assumptions on r and d, the upper and lower bounds agree on both type-1 and type-2 arbitrary triangulations. In addition, a dimension criterion which guarantees that the lower bound gives the actual dimension is given.

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  • [1] C. K. Chui, Multivariate Splines, CBMS-NSF Lecture Series in Applied Math., no. 54, SIAM, Philadelphia, PA, 1988. MR 1033490 (92e:41009)
  • [2] C. K. Chui & M. Lai, "On multivariate vertex splines and applications," in 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)
  • [3] C. K. Chui & R. H. Wang, "Multivariate spline spaces," J. Math. Anal. Appl., v. 94, 1983, pp. 197-221. MR 701458 (84f:41010)
  • [4] R. H. J. Gmelig Meyling & P. R. Pfluger, "On the dimension of the spline space $ S_2^1(\Delta )$ in special cases," in Multivariate Approximation Theory III (W. Schempp and K. Zeller, eds.), Birkhäuser, Basel, 1985, pp. 180-190.
  • [5] J. Morgan & R. Scott, "A nodal basis for $ {C^1}$ piecewise polynomials in two variables," Math. Comp., v. 29, 1975, pp. 736-740. MR 0375740 (51:11930)
  • [6] L. L. Schumaker, "On the dimension of spaces of piecewise polynomials in two variables," in Multivariate Approximation Theory (W. Schempp and K. Zeller, eds.), Birkhäuser, Basel, 1979, pp. 396-412. MR 560683 (81d:41011)
  • [7] L. L. Schumaker, "Bounds on the dimension of spaces of multivariate piecewise polynomials," Rocky Mountain J. Math., v. 14, 1984, pp. 251-264. MR 736177 (85h:41091)
  • [8] L. L. Schumaker, "On super splines and finite elements," SIAM J. Numer. Anal. (To appear.) MR 1005521 (90g:65016)
  • [9] A. Ženišek, "Polynomial approximation on tetrahedrons in the finite element method," J. Approx. Theory, v. 7, 1973, pp. 334-351. MR 0350260 (50:2753)
  • [10] A. Ženišek, "A general theorem on triangular $ {c^m}$ elements," RAIRO Anal. Numér., v. 22, 1974, pp. 119-127.

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Keywords: Bivariate splines, dimension, supersplines, lower and upper bound, quasi-crosscut partition, type-1 and type-2 triangulations, dimension criterion
Article copyright: © Copyright 1989 American Mathematical Society

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