A nodal basis for the $C^1$-$P_{33}$ finite elements on 5D simplex grids

TL;DR

This paper constructs a specialized nodal basis for a high-degree, smooth finite element space on 5D simplex grids, enabling advanced numerical analysis in higher dimensions.

Contribution

It introduces a novel nodal basis for the $C^1$ finite element space of degree 33 on 5D simplices, with detailed smoothness properties at various simplex features.

Findings

Provides explicit basis construction for 5D $C^1$ finite elements.

Ensures high smoothness levels at faces, edges, and vertices.

Facilitates higher-dimensional finite element analysis.

Abstract

We construct a nodal basis for the 5-dimensional $C^1$ finite element space of polynomial degree $33$ on simplex grids, where the finite element functions are $C^1$ on the 6 4D-simplex faces, $C^2$ on the 15 face-tetrahedra, $C^4$ on the 20 face-triangles, $C^8$ on the 15 edges, and $C^{16}$ at the 6 vertices, of a 5D simplex.