A characterization of supersmoothness of multivariate splines

TL;DR

This paper characterizes supersmoothness in multivariate splines over simplicial meshes, linking it to polynomial spline degeneracy and determining maximal supersmoothness orders for different cell configurations.

Contribution

It provides a new characterization of supersmoothness via polynomial spline degeneracy and identifies maximal supersmoothness orders for various configurations.

Findings

Supersmoothness relates to polynomial spline space degeneracy.

Maximal supersmoothness orders depend on cell configuration.

Characterization aids in multivariate spline construction and finite element methods.

Abstract

We consider spline functions over simplicial meshes in $\RR^n$. We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as \emph{supersmoothness}, which plays a role in the construction of multivariate splines and in the finite element method. In this paper we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations.