\(C^ r\)-finite elements of Powell-Sabin type on the three direction mesh
The triangulation \(\tau\) generated by a uniform three direction mesh of the plane and the Powell-Sabin subtriangulation \(\tau_6\) obtained by subdividing each triangle \(T\in\tau\) by connecting each vertex to the midpoint of the opposite side are considered. The existence of Hermite interpolation schemes in a subspace of \(S^r_n (\tau_6)\) for lower degrees (i.e. \(n=2r+1\) for \(r\) even and \(n=2r\) for \(r\) odd) is proven. Some results on the Bernstein-Bézier form of polynomials on triangles which is used for representing splines on the triangulation \(\tau_6\) are given. The construction of Powell-Sabin finite elements and the solution of the Hermite interpolation problem of order \(r\) are discussed. The interpolation error is estimated.
scientific article; zbMATH DE number 989553