## On the dimension of bivariate superspline spaces

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- by Charles K. Chui and Tian Xiao He PDF
- Math. Comp.
**53**(1989), 219-234 Request permission

## Abstract:

A bivariate piecewise polynomial function of total degree*d*on some grid partition $\Delta$ that has

*r*th 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.

## References

- Charles K. Chui,
*Multivariate splines*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. MR**1033490**, DOI 10.1137/1.9781611970173 - Charles K. Chui and Ming Jun Lai,
*On multivariate vertex splines and applications*, Topics in multivariate approximation (Santiago, 1986) Academic Press, Boston, MA, 1987, pp. 19–36. MR**924820** - Charles K. Chui and Ren Hong Wang,
*Multivariate spline spaces*, J. Math. Anal. Appl.**94**(1983), no. 1, 197–221. MR**701458**, DOI 10.1016/0022-247X(83)90014-8
R. H. J. Gmelig Meyling & P. R. Pfluger, "On the dimension of the spline space $S_2^1(\Delta )$ in special cases," in - John Morgan and Ridgway Scott,
*A nodal basis for $C^{1}$ piecewise polynomials of degree $n\geq 5$*, Math. Comput.**29**(1975), 736–740. MR**0375740**, DOI 10.1090/S0025-5718-1975-0375740-7 - Larry L. Schumaker,
*On the dimension of spaces of piecewise polynomials in two variables*, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel-Boston, Mass., 1979, pp. 396–412. MR**560683** - Larry L. Schumaker,
*Bounds on the dimension of spaces of multivariate piecewise polynomials*, Rocky Mountain J. Math.**14**(1984), no. 1, 251–264. Surfaces (Stanford, Calif., 1982). MR**736177**, DOI 10.1216/RMJ-1984-14-1-251 - Larry L. Schumaker,
*On super splines and finite elements*, SIAM J. Numer. Anal.**26**(1989), no. 4, 997–1005. MR**1005521**, DOI 10.1137/0726055 - Alexander Ženíšek,
*Polynomial approximation on tetrahedrons in the finite element method*, J. Approximation Theory**7**(1973), 334–351. MR**350260**, DOI 10.1016/0021-9045(73)90036-1
A. Ženišek, "A general theorem on triangular ${c^m}$ elements,"

*Multivariate Approximation Theory*III (W. Schempp and K. Zeller, eds.), Birkhäuser, Basel, 1985, pp. 180-190.

*RAIRO Anal. Numér.*, v. 22, 1974, pp. 119-127.

## Additional Information

- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp.
**53**(1989), 219-234 - MSC: Primary 41A15
- DOI: https://doi.org/10.1090/S0025-5718-1989-0969483-6
- MathSciNet review: 969483