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Tiniest tensor H(div)

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Alternative namesTNT H(div)
Cockburn–fu names\(\left[S_{3,k}^\square\right]_{d-1}\)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(3)}_{k}\) (quadrilateral)
\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(4)}_{k}\) (hexahedron)
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DOFsOn each facet: normal integral moments with an order \(k\) Lagrange space
On the interior of the reference element: integral moments with \(\nabla f\) for each \(f\) in an order \(k\) Lagrange space, and integral moments with \(\nabla\times\boldsymbol{f}\) for each \(\boldsymbol{f}\) in an order \(k\) vector Lagrange space such that the tangential trace of \(\boldsymbol{f}\) on the facets of the cell is 0
Number of DOFsquadrilateral: \(2(k+1)^2 + 3\)
hexahedron: \(3(k+1)^3 + 7\)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"TNTdiv" (quadrilateral, hexahedron)
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Examples

quadrilateral
order 1

(click to view basis functions)
quadrilateral
order 2

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quadrilateral
order 3

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hexahedron
order 1

(click to view basis functions)

References

DefElement stats

Element added24 October 2021
Element last updated02 August 2022