Serendipity H(div)

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Alternative names	Brezzi–Douglas–Marini cubical H(div) (quadrilateral), Arnold–Awanou H(div) (hexahedron)
De Rham complex families	\(\left[S_{1,k}^\square\right]_{d-1}\) or \(\mathcal{S}_{k}\Lambda^{d-1}(\square_d)\)
Abbreviated names	BDMcf (quadrilateral), AAf (hexahedron)
Degrees	\(1\leqslant k\) where \(k\) is the Polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Lagrange subdegree	\(\operatorname{floor}(k/d)\)
Lagrange superdegree	\(k+1\)
Reference elements	quadrilateral, hexahedron
Polynomial set	\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(28)}_{k}\) (quadrilateral) \(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(29)}_{k}\) (hexahedron) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(28)}_k=\operatorname{span}\left\{\left(\begin{array}{c}x^{k+1}\\(k+1)x^ky\end{array}\right),\left(\begin{array}{c}(k+1)xy^k\\y^{k+1}\end{array}\right)\right\}\) \(\mathcal{Z}^{(29)}_k=\left\{\nabla\times\boldsymbol{p}\middle\|\boldsymbol{p}\in\mathcal{A}_{k}\right\}\) \(\mathcal{A}_k=\operatorname{span}\bigcup_{p\in\mathcal{P}_k(\mathbb{R}^{d-1})}\left\{\left(\begin{array}{c}0\\xzp(y,z)\\-xyp(y,z)\end{array}\right),\left(\begin{array}{c}yzp(x,z)\\0\\-xyp(x,z)\end{array}\right),\left(\begin{array}{c}yzp(x,y)\\-xzp(x,y)\\0\end{array}\right)\right\}\)
DOFs	On each facet: normal integral moments with an degree \(k\) dPc space On the interior of the reference element: integral moments with an degree \(k-2\) vector dPc space
Number of DOFs	quadrilateral: \(k^2+3k+4\) (A014206) hexahedron: \((k+1)(k^2+5k+12)/2\)
Mapping	contravariant Piola
continuity	Components normal to facets are continuous
Categories	Vector-valued elements, H(div) conforming elements

Implementations

This element is implemented in Symfem and (legacy) UFL.↓ Show implementation detail ↓

Examples

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

References

Arnold, Douglas N. and Awanou, Gerard. Finite element differential forms on cubical meshes, Mathematics of computation 83, 1551–5170, 2014. [BibTeX]
Brezzi, Franco, Douglas, Jim, and Marini, L. Donatella. Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik 47, 217–235, 1985. [DOI: 10.1007/BF01389710] [BibTeX]
Arnold, Douglas N. and Logg, Anders. Periodic table of the finite elements, SIAM News 47, 2014. [www.siam.org/publications/siam-news/issues/volume-47-number-09-november-2014/] [BibTeX]
Cockburn, Bernardo and Fu, Guosheng. A systematic construction of finite element commuting exact sequences, SIAM journal on numerical analysis 55, 1650–1688, 2017. [DOI: 10.1137/16M1073352] [BibTeX]

DefElement stats

Element added	30 December 2020
Element last updated	27 September 2024