Construction of Local $C^1$ 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 $S_4^1(\Delta )$ 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.