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Arnold–Boffi–Falk

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Abbreviated names	ABF
Degrees	\(0\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(2k+2\)
Lagrange subdegree	\(k\)
Lagrange superdegree	\(k+2\)
Reference cells	quadrilateral
Finite dimensional space	\(\mathcal{Z}^{(0)}_{k} \oplus \mathcal{Z}^{(1)}_{k}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{Z}^{(0)}_k=\left\{\left(\begin{array}{c}x^py^q\\0\end{array}\right)\middle\|p\leqslant k+2,q\leqslant k\right\}\) \(\mathcal{Z}^{(1)}_k=\left\{\left(\begin{array}{c}0\\x^py^q\end{array}\right)\middle\|p\leqslant k,q\leqslant k+2\right\}\)
DOFs	On each edge: normal integral moments with a degree \(k\) Lagrange space On each face: integral moments with a degree \(k\) Nédélec (first kind) space, integral moments of the divergence with \(x^{k+1}y^q\) for q=0,1,...,k, and integral moments of the divergence with \(x^qy^{k+1}\) for q=0,1,...,k
Mapping	see [2]
continuity	Components normal to facets are continuous
Categories	Vector-valued elements, H(div) conforming elements

Implementations

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Examples

quadrilateral degree 0	(click to view basis functions)
quadrilateral degree 1	(click to view basis functions)
quadrilateral degree 2	(click to view basis functions)

References

[1] Arnold, Douglas N., Boffi, Daniele, and Falk, Richard S. Quadrilateral H(div) finite elements, SIAM Journal on Numerical Analysis 42(5), 2429–2451, 2005. [DOI: 10.1137/S0036142903431924] [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]

DefElement stats

Element added	01 December 2021
Element last updated	20 November 2025