Construction of local C1 quartic spline elements for optimal-order approximation

This paper is concerned with a study of approximation order and construction of locally supported elements for the space S 1 4 (() of C 1 pp (piecewise polynomial) functions on an arbitrary triangulation of a connected polygonal domain in R 2. It is well-known that even when is a three-directional mesh (1) , the order of approximation of S 1 4 (((1)) is only 4, not 5. The objective of this paper is twofold: (i) A local Clough-Tocher reenement procedure of an arbitrary triangulation is introduced so as to yield the optimal ((fth) order of approximation, where locality means that only a few isolated triangles need reenement, and (ii) locally supported Hermite elements are constructed to achieve the optimal order of approximation.