A family of C1 Clough–Tocher spline spaces on C0 piecewise quadratic domain partitions

The paper addresses the construction of $C^1$ splines on a curved domain that is parametrized by a $C^0$ piecewise geometry mapping composed of quadratic Bézier triangles. The $C^1$ splines are assembled from polynomials of a chosen total degree greater than or equal to four, and their construction is based on the Clough–Tocher splitting technique that ensures locality. In particular, the splines are locally characterized by an interpolation problem described by Hermite data, which resembles the standard macro-element concepts developed for $C^1$ splines on triangulations.