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Transactions of the American Mathematical Society

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Stability of optimal-order approximation by bivariate splines over arbitrary triangulations

Authors: C. K. Chui, D. Hong and R. Q. Jia
Journal: Trans. Amer. Math. Soc. 347 (1995), 3301-3318
MSC: Primary 41A15; Secondary 41A63, 65D07
MathSciNet review: 1311906
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Abstract: Let $ \Delta $ be a triangulation of some polygonal domain in $ {\mathbb{R}^2}$ and $ S_k^r(\Delta )$, the space of all bivariate $ {C^r}$ piecewise polynomials of total degree $ \leqslant k$ on $ \Delta $. In this paper, we construct a local basis of some subspace of the space $ S_k^r(\Delta )$, where $ k \geqslant 3r + 2$, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of $ \Delta $ with the exception of the smallest angle in the partition. This result is obtained by means of a careful choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimal-order approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their $ {\text{B}}$-net representations is derived for this purpose.

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Keywords: Bivariate splines, triangulations, $ {\text{B}}$-net representations, approximation order, local bases, stability
Article copyright: © Copyright 1995 American Mathematical Society

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