A Construction of $C^{r}$ Conforming Finite Elements on the Alfeld Split in Any Dimension
This addresses a long-standing issue in numerical analysis for researchers and practitioners, offering incremental improvements over recent methods.
The paper tackles the problem of constructing C^r conforming finite element spaces in any dimension by providing a first unified construction on the Alfeld split, relaxing supersmoothness conditions and polynomial degree requirements compared to prior work.
Constructing $C^r$ conforming finite element spaces in any dimension is a long-standing problem. For general triangulations, this problem was recently addressed by Hu-Lin-Wu (2024), under certain conditions on supersmoothness and polynomial degree. In this paper, a first unified construction on the Alfeld split in any dimension is given, where the supersmoothness conditions and the polynomial degree requirement are relaxed.