The sharpness condition for constructing a finite element from a superspline

TL;DR

This paper establishes sharpness conditions for constructing $C^r$ finite element spaces from superspline spaces on simplicial triangulations, highlighting the necessary continuity and polynomial degree requirements.

Contribution

It introduces the extendability concept for pre-element spaces and provides conditions for constructing $C^r$ conforming finite elements from supersplines.

Findings

Constructing $C^r$ elements requires extra $C^{2^{s}r}$ continuity on $s$-simplices.

Polynomial degree must be at least $(2^d r + 1)$.

Extendability condition unifies superspline and finite element space construction.

Abstract

This paper addresses sharpness conditions for constructing $C^r$ conforming finite element spaces from a superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre-element spaces, which encompasses both the superspline spaces and the finite element spaces. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing $C^r$ conforming elements in $d$ dimensions generally requires an extra $C^{2^{s}r}$ continuity on $s$-codimensional simplices, and the polynomial degree is at least $(2^d r + 1)$.