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Construction of Local $C^1$ Quartic Spline
Elements for Optimal-Order Approximation


Authors: Charles K. Chui and Dong Hong
Journal: Math. Comp. 65 (1996), 85-98
MSC (1991): Primary 41A25, 41A63; Secondary 41A05, 41A15, 65D07
DOI: https://doi.org/10.1090/S0025-5718-96-00689-8
MathSciNet review: 1325865
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Abstract: This paper is concerned with a study of approximation order and construction of locally supported elements for the space $S_4^1(\Delta )$ of $C^1$ $pp$ (piecewise polynomial) functions on an arbitrary triangulation $\Delta $ of a connected polygonal domain $\Omega $ in $\mathbb{R}^2$. It is well known that even when $\Delta $ is a three-directional mesh $\Delta ^{(1)}$, the order of approximation of $S_4^1(\Delta ^{(1)})$ is only 4, not 5. The objective of this paper is two-fold: (i) A local Clough-Tocher refinement procedure of an arbitrary triangulation $\Delta $ is introduced so as to yield the optimal (fifth) order of approximation, where locality means that only a few isolated triangles need refinement, and (ii) locally supported Hermite elements are constructed to achieve the optimal order of approximation.


References [Enhancements On Off] (What's this?)

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Additional Information

Charles K. Chui
Affiliation: Center for Approximation Theory, Texas A&M University, College Station, Texas 77843
Email: cchui@tamu.edu

Dong Hong
Affiliation: Center for Approximation Theory, Texas A&M University, College Station, Texas 77843
Email: dhong@math.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-96-00689-8
Keywords: Approximation order, B-net representations, bivariate splines, local Clough-Tocher refinement, star-vertex splines, triangulations
Received by editor(s): May 28, 1994
Received by editor(s) in revised form: December 5, 1994
Additional Notes: Research supported by NSF Grant No. DMS 92-06928 and ARO Contract DAAH 04-93-G-0047
Article copyright: © Copyright 1996 American Mathematical Society

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