Abstract
This paper is concerned with a study of approximation order and construction of locally supported elements for the space S 4 1 ( Π) S_4^1(\Delta ) of C 1 C^1 p p pp (piecewise polynomial) functions on an arbitrary triangulation Π\Delta of a connected polygonal domain Ω \Omega in R 2 \Bbb R^2 . It is well known that even when Π\Delta is a three-directional mesh Π( 1 ) \Delta ^{(1)} , the order of approximation of S 4 1 ( Π( 1 ) ) 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 Π\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.
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