The sharpness condition for constructing a finite element from a superspline

Hu, Jun; Lin, Ting; Wu, Qingyu; Yuan, Beihui

doi:10.1090/mcom/4119

The sharpness condition for constructing a finite element from a superspline
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by Jun Hu, Ting Lin, Qingyu Wu and Beihui Yuan;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4119

Published electronically: June 26, 2025

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Abstract:

This paper addresses sharpness conditions for constructing $C^r$ conforming finite element spaces from 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 $C^{2^{s - 1} r}$ continuity on $s$-codimensional subsimplices, and the polynomial degree is at least $(2^d r + 1)$.

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The sharpness condition for constructing a finite element from a superspline
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Abstract: