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Guzmán–Neilan (first kind)

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Alternative names	Guzmán–Neilan
Abbreviated names	GN
Degrees	$1 ⩽ k ⩽ d - 1$ where $k$ is the Polynomial subdegree
Polynomial subdegree	$k$
Polynomial superdegree	$d$
Reference elements	triangle, tetrahedron
DOFs	On each vertex: point evaluations in coordinate direcions On each edge: (if $k > 1$ ) point evaluations in coordinate directions at midpoints On each facet: normal integral moments with a degree $0$ Lagrange space
Number of DOFs	triangle: $9$ tetrahedron: ${\begin{cases} 16 & k = 1 \\ 34 & k = 2 \end{cases}$
Mapping	contravariant Piola
continuity	Function values are continuous.
Notes	This element is a modification of the Bernardi–Raugel element with the facet bubbles modified to be divergence free.
Categories	Vector-valued elements, Macro elements

Implementations

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

Examples

triangle degree 1	(click to view basis functions)
tetrahedron degree 1	(click to view basis functions)
tetrahedron degree 2	(click to view basis functions)

References

Guzmán, Johnny and Neilan, Michael. Inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions, SIAM Journal on Numerical Analysis 56, 2826–2844, 2018. [DOI: 10.1137/17M1153467] [BibTeX]

DefElement stats

Element added	01 August 2021
Element last updated	17 October 2024