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

TL;DR

This paper constructs and proves the properties of $C^m$-$P_k^{(n)}$ finite element spaces on 3D and 4D simplicial grids, providing explicit bases and a code for their generation.

Contribution

It introduces a method to construct nodal bases for $C^m$-$P_k^{(n)}$ finite elements on tetrahedral and 4D grids, including proofs of unisolvency and continuity.

Findings

Constructed nodal bases for $C^m$-$P_k^{(3)}$ and $C^m$-$P_k^{(4)}$.

Proved unisolvency and $C^m$ continuity of these finite element spaces.

Provided a computer code for generating the basis index sets.

Abstract

We construct the nodal basis of $C^m$-$P_{k}^{(3)}$ ($k \ge 2^3m+1$) and $C^m$-$P_{k}^{(4)}$ ($k \ge 2^4m+1$) finite elements on 3D tetrahedral and 4D simplicial grids, respectively. $C^m$-$P_{k}^{(n)}$ stands for the space of globally $C^m$ ($m\ge1$) 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.