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

DOI:
https://doi.org/10.1090/S0002-9947-1995-1311906-6

MathSciNet review:
1311906

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Abstract: Let be a triangulation of some polygonal domain in and , the space of all bivariate piecewise polynomials of total degree on . In this paper, we construct a local basis of some subspace of the space , where , 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 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 -net representations is derived for this purpose.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1311906-6

Keywords:
Bivariate splines,
triangulations,
-net representations,
approximation order,
local bases,
stability

Article copyright:
© Copyright 1995
American Mathematical Society