Locally generated $\mathcal{C}^1$-splines over triangular meshes

Abstract

We classify all possible local linear procedures over triangular meshes resulting in polynomial C 1 -spline functions with affinely uniform shape for the basic functions at the edges, and fitting the 9 value-and gradient data at the vertices of the mesh members. There is a unique procedure among them with shape functions and basic polynomials of degree 5 and all other admissible procedures are its pertubations with higher degree.

A Simple Affine-Invariant Spline Interpolation over Triangular Meshes

“…Originally, they only proved that the linear system of 21 equations for calculating the 21 coefficients for the adjustment admits a unique solution. Recently, Sergienko et al (2014) [4] published the rather sophisticated related explicit formulas, which motivated us to develop an axiomatic approach to locally generated polynomial spline methods Stachó (2019) [5] The recent work is a non-straightforward application of the results there, although it is self-contained formally. We only used the principal shape functions Φ and Θ below provided by Theorem 2.3 in Stachó (2019) [5] in the simplest form without the need for any hint of their provenience.…”