zbMATH — the first resource for mathematics

Multivariate vertex splines and finite elements. (English) Zbl 0702.41017
In this interesting paper the authors present tremendous theorems, examples of vertex splines and the pictures of these splines, therefore we cannot review them; we recall only the summary: “The objective of this paper is to present a unified study of multivariate super vertex splines with emphasis on the construction procedure and an application to least-square approximation with interpolatory constraints. Both simplicial and parallelepiped partitions are studied in some detail, and in the bivariate setting, even a partition consisting of both triangles and parallelograms is considered. When the polynomial degree is allowed to be sufficiently large as compared to the order of smoothness, it is clear that vertex splines can be constructed by working on each simplex or parallelepiped separately as long as certain suitable normal derivative constraints are imposed on the boundary faces. Our constructive procedure will take a different route. Instead of normal derivatives, we impose extra interpolatory conditions at the “vertices”. This gives rise to the notion of “super splines” introduced in this paper. It should also be emphasized that the view point of considering a basis of piecewise polynomials with smallest possible supports so that the full approximation order is preserved makes vertex splines different from the standard approach in finite elements. After all, if the polynomial degree is required to be lower, it is necessary to work on at least three adjacent simplices or parallelepipeds simultaneously in constructing a basis of vertex splines”.
Reviewer: L.Leindler

41A15 Spline approximation
Full Text: DOI
[1] Alfeld, P; Schumaker, L.L, The dimension of bivariate spline spaces of smoothness r for degree d⩾4r + 1, Constr. approx., 3, 189-197, (1987) · Zbl 0646.41008
[2] Bézier, P, Numerical control: mathematics and applications, (1972), Wiley London · Zbl 0251.93002
[3] Böhm, W; Farin, G; Kahmann, J, A survey of curve and surface methods in CAGD, Comput. aided geom. design, 1, 1-60, (1984) · Zbl 0604.65005
[4] de Boor, C, A practical guide to splines, (1978), Springer Verlag New York/Berlin · Zbl 0406.41003
[5] de Boor, C, B-form basics, (), 131-148
[6] Bramble, J.H; Hilbert, S.R, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. math., 16, 362-369, (1971) · Zbl 0214.41405
[7] de Casteljau, P, Courbes et surfaces à poles, (1959), André Citroën Automobiles, S. A Paris
[8] Chui, C.K, Multi-dimensional spline techniques for Fitting of surfaces to wind fields over complex terrain, Final report, (August 1986), Battelle
[9] Chui, C.K, Multivariate splines, () · Zbl 0619.41004
[10] Chui, C.K; He, T.X, On the dimension of bivariate super spline spaces, CAT report no. 141, (April, 1987)
[11] Chui, C.K; Lai, M.J, On bivariate vertex splines, (), 84-115 · Zbl 0588.65009
[12] Chui, C.K; Lai, M.J, On multivariate vertex splines and applications, (), 19-36 · Zbl 0588.65009
[13] Chui, C.K; Lai, M.J, On bivariate super vertex splines, CAT report no. 164, (1988) · Zbl 0588.65009
[14] Dahmen, W, Bernstein-Bézier representation of polynomial surfaces, (), organized by C. de Boor, Dallas, Texas · Zbl 0606.41009
[15] Farin, G, Subsplines über dreiecken, Dissertation, (1979), Braunschweig
[16] Farin, G, Triangular Bernstein-Bézier patches, Comput. aided geom. design, 3, 83-127, (1986)
[17] Gregory, J.A, Interpolation to boundary data on the simplex, Comput. aided geom. design, 2, 43-52, (1985) · Zbl 0582.65004
[18] Hayes, J.G, Some practical applications in discrete multivariate approximation, () · Zbl 0415.41019
[19] Jia, R.Q, B-net representation of multivariate splines, Kexue tongbao, 11, 804-807, (1987), [Chinese]
[20] Lai, M.J, Construction of bivariate and trivariate vertex splines on arbitrary mixed grid partitions, ()
[21] Le Méhauté, A, Interpolation et approximation par des fonctions polynomials par morceaux dans rn, ()
[22] Le Méhauté, A, Unisolvent interpolation in rn and simplicial polynomial finite elements methods, (), 141-151
[23] Luenberger, D.G, Optimization by vector space methods, (1969), Wiley New York · Zbl 0176.12701
[24] Sablonniére, P, Composite finite elements of C2, (), 207-217
[25] Schoenberg, I.J, Cardinal spline interpolation, () · Zbl 0264.41003
[26] Schultz, M.H, Discrete Tchebycheff approximation for multivariate splines, J. comput. system sci., 6, 298-304, (1972) · Zbl 0244.41001
[27] Schumaker, L.L, Spline functions: basic theory, (1981), Wiley New York · Zbl 0449.41004
[28] Schumaker, L.L, Triangulation methods, (), 219-232 · Zbl 0632.65120
[29] Z̆enisĕk, A, Interpolation polynomials on the triangle, Numer. math., 15, 283-296, (1970) · Zbl 0216.38901
[30] Z̆enisĕk, A, Polynomial approximation on tetrahedrons in the finite element method, J. approx. theory, 7, 334-351, (1973) · Zbl 0279.41005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.