A bivariate \(C^ 3\) interpolation scheme (Q1903659)

On the twice Clough-Tochter type split of a triangle a \(C^3\) scheme is constructed. This split has the minimal number of subtriangles needed for the existence of a \(C^3\) scheme. It is proved that there exists a \(C^3\) piecewise polynomial of degree 7 interpolating arbitrarily given values and derivatives of orders up to 3 at the vertices and on the edges of the triangle. The discussed method is a mixture of \(B\)-forms of polynomials and the serendipity of bivariate splines.