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.

的空间$C^{\mathrm{1个}}$三次 Clough–Tocher 样条是用于求解偏微分方程的三角剖分上的经典有限元近似空间。然而，对于这样的空间，没有可用的 B 样条基，而 B 样条基是计算机辅助几何设计和等几何分析的首选。B 样条基是形成统一凸分区的局部支持基。在本文中，我们探索了几种替代方案$C^{\mathrm{1个}}$配备 B 样条基的三角剖分上的三次样条空间。它们是在 Powell-Sabin 精细三角剖分上定义的，并呈现不同类型的$C^{\mathrm{2个}}$超顺滑。超平滑B样条是通过提取过程获得的，即，它们用不太平滑的基函数表示。这些替代样条空间保持与 Clough-Tocher 样条相同的最佳逼近能力。在二阶和四阶边界值问题的最小二乘近似和有限元近似的背景下，通过选择数值示例来说明这一点。