Optimal-order approximation by mixed three-directional spline elements

This paper is concerned with a study of approximation order and construction of locally supported elements for the space S14(Δ) of C1 quartic pp (piecewise polynomial) functions on a triangulation Δ of a connected polygonal domain Ω in R2. It is well known that, when Δ is a three-directional mesh Δ(1), the order of approximation of S14(Δ(1)) is only 4, not 5. Though a local Clough-Tocher refinement procedure of an arbitrary triangulation Δ yields the optimal (fifth) order of approximation from the space S14(Δ) (see [1]), it needs more data points in addition to the vertex set of the triangulation Δ. In this paper, we will introduce a particular mixed three-directional mesh Δ(3) and construct so-called mixed three-directional elements. We prove that the space S14(Δ(3)) achieves its optimal-order of approximation by constructing an interpolation scheme using mixed three-directional elements.