The space of $C^1$ cubic Clough-Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative $C^1$ cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell-Sabin refined triangulation and present different types of $C^2$ super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough-Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems.

翻译：$C^1$ 三次Clough-Tocher样条空间是用于求解偏微分方程的三角剖分上的经典有限元逼近空间。然而，该空间缺乏B样条基函数，而B样条基是计算机辅助几何设计与等几何分析中的首选。B样条基是一组具有局部支撑性且构成凸单位分割的基函数。本文探索了若干种定义在三角剖分上且具备B样条基的替代性$C^1$ 三次样条空间。这些空间建立在Powell-Sabin细化三角剖分上，并展现出不同类型的$C^2$ 超光滑性。超光滑B样条通过提取过程获得，即将其表示为较不光滑基函数的线性组合。这些替代样条空间保持了与Clough-Tocher样条相同的最优逼近能力。本文通过最小二乘逼近及针对二阶与四阶边值问题的有限元逼近中的一系列数值算例验证了该结论。