Bell

Degrees	$k=4$ where $k$ is the polynomial subdegree
Polynomial subdegree	$k$
Polynomial superdegree	$k+1$
Reference cells	triangle
Finite dimensional space	$\mathcal{Z}^{(4)}_{k+1}$ ↓ Show set definitions ↓↑ Hide set definitions ↑ $\mathcal{Z}^{(4)}_k=\left\{p\in\mathcal{P}_{k}\middle\|\text{normal derivative of $p$ on each edge is cubic}\right\}$ $\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}$
DOFs	On each vertex: point evaluations, point evaluations of first derivatives, and point evaluations of second derivatives
Number of DOFs	triangle: $18$
Mapping	see [2]
continuity	Function values and first and second derivatives are continuous.
Categories	Scalar-valued elements

Implementations

This element is implemented in FIAT and Symfem .↓ Show implementation detail ↓

triangle
degree 4

[1] Bell, Kolbein. A refined triangular plate bending finite element, International Journal for Numerical Methods in Engineering 1(1), 101–122, 1969. [DOI: 10.1002/nme.1620010108] [BibTeX]
[2] Kirby, Robert C. A general approach to transforming finite elements, The SMAI journal of computational mathematics 4, 197–224, 2018. [DOI: 10.5802/smai-jcm.33] [BibTeX]
[3] Kirby, Robert C. A general approach to transforming finite elements, SMAI Journal of Computational Mathematics 4, 197–224, 2018. [DOI: 10.5802/smai-jcm.33] [BibTeX]

Element added	20 February 2021
Element last updated	20 November 2025

Degrees	\(k=4\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Reference cells	triangle
Finite dimensional space	\(\mathcal{Z}^{(4)}_{k+1}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{Z}^{(4)}_k=\left\{p\in\mathcal{P}_{k}\middle\|\text{normal derivative of $p$ on each edge is cubic}\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On each vertex: point evaluations, point evaluations of first derivatives, and point evaluations of second derivatives
Number of DOFs	triangle: \(18\)
Mapping	see [2]
continuity	Function values and first and second derivatives are continuous.
Categories	Scalar-valued elements