In this thesis we present the Clough-Tocher space of cubic splines on a triangulation, where we make a refinement of the triangulation by splitting each triangle on three smaller ones. The spline is composed of cubic polynomials over each triangle and it is ${\cal C}^1$ over the whole triangulation. We focus on a special subspace - the reduced Clough-Tocher spline space and construct a normalized basis for it. The basis splines have a local support, they are nonnegative and they form a partition of unity. Geometrically, the basis construction problem is converted to a problem of finding a set of triangles that contain specific points. This leads us to control triangles and a stable way of computing with Clough-Tocher splines.
|
