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arXiv · 2202.05837

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

Shangyou Zhang
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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.

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Shangyou Zhang. 2022-02-10. The nodal basis of $C^m$-$P_{k}^{(3)}$ and $C^m$-$P_{k}^{(4)}$ finite elements on tetrahedral and 4D simplicial grids. https://arxiv.org/abs/2202.05837

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