Optimal Lagrange interpolation by quartic C1C1 splines on triangulations

We develop a local Lagrange interpolation scheme for quartic C1C1 splines on triangulations. Given an arbitrary triangulation ΔΔ, we decompose ΔΔ into pairs of neighboring triangles and add “diagonals” to some of these pairs. Only in exceptional cases, a few triangles are split. Based on this simple refinement of ΔΔ, we describe an algorithm for constructing Lagrange interpolation points such that the interpolation method is local, stable and has optimal approximation order. The complexity for computing the interpolating splines is linear in the number of triangles. For the local Lagrange interpolation methods known in the literature, about half of the triangles have to be split.