Simplex-splines on the Clough–Tocher split for high-order numerical approximation

Smooth splines defined on (possibly refined) triangulations are widely used in the area of data fitting and numerical analysis of partial differential equations, in particular isogeometric analysis. A classical choice is the C1 cubic spline space defined on the Clough-Tocher split, which divides each triangle into three subtriangles by connecting the barycenter of the triangle to its three vertices. In this paper, we consider the more general family of C1 spline spaces of degree d >= 3. Recently, a simplex-spline basis was developed with the aim of having an effective local representation of such splines on each triangle. Here we investigate the use of these (local) simplex-splines as a practical tool to build C1 spline approximations over general triangulations and numerically test them in the least-squares method for function approximation and in the Galerkin method for solving linear and semilinear elliptic problems.