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

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.