Stability of optimal-order approximation by bivariate splines over arbitrary triangulations
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- by C. K. Chui, D. Hong and R. Q. Jia PDF
- Trans. Amer. Math. Soc. 347 (1995), 3301-3318 Request permission
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.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3301-3318
- MSC: Primary 41A15; Secondary 41A63, 65D07
- DOI: https://doi.org/10.1090/S0002-9947-1995-1311906-6
- MathSciNet review: 1311906