Extraction and application of super-smooth cubic B-splines over triangulations

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