The nodal basis of $C^m$-$P_k^{(3)}$ and $C^m$-$P_k^{(4)}$ finite elements on tetrahedral and 4D simplicial grids

Abstract

We construct the nodal basis of $C^m$-$P_k^{(3)}$ ($k\ge 2^{3}m+1$) and $C^m$-$P_k^{(4)}$ ($k\ge 2^{4}m+1$) finite elements on 3D tetrahedral and 4D simplicial grids, respectively. $C^m$-$P_k^{(n)}$ stands for the space of globally $C^m$ ($m\ge 1$) and locally piecewise $n$-dimensional polynomials of degree $k$ on $n$-dimensional simplicial grids. We prove the uni-solvency and the $C^m$ continuity of the constructed $C^m$-$P_k^{(3)}$ and $C^m$-$P_k^{(4)}$ finite element spaces. A computer code is provided which generates the index set for the nodal basis of $C^m$-$P_k^{(n)}$ finite elements on $n$-dimensional simplicial grids.