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Hellan–Herrmann–Johnson

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Degrees\(0\leqslant k\)
where \(k\) is the Polynomial subdegree
Polynomial subdegree\(k\)
Polynomial superdegree\(k\)
Reference cellstriangle, tetrahedron
Polynomial set\(\mathcal{Z}^{(2)}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn each facet: integral moments of inner products of normal with a degree \(k\) Lagrange space
On the interior of the reference cell (triangle): integrals against tensor products with symmetric matrices whose entries are in a degree \(k-1\) Lagrange space
On the interior of the reference cell (tetrahedron): integrals against tensor products with symmetric matrices whose entries are in a degree \(k-1\) Lagrange space, and integrals against tensor products with the matrices \(\frac12(\boldsymbol{n}_1\boldsymbol{n}_2^{\text{t}}+\boldsymbol{n}_2\boldsymbol{n}_1^{\text{t}})\) and \(\frac12(\boldsymbol{n}_2\boldsymbol{n}_3^{\text{t}}+\boldsymbol{n}_3\boldsymbol{n}_2^{\text{t}})\) multiplied by degree \(k\) polynomials
Number of DOFstriangle: \(3(k+1)(k+2)/2\) (A045943)
Mappingdouble contravariant Piola
continuityInner products with normals to facets are continuous
CategoriesMatrix-valued elements

Implementations

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

Examples

triangle
degree 0

(click to view basis functions)
triangle
degree 1

(click to view basis functions)
triangle
degree 2

(click to view basis functions)
tetrahedron
degree 0

(click to view basis functions)
tetrahedron
degree 1

(click to view basis functions)
tetrahedron
degree 2

(click to view basis functions)

References

DefElement stats

Element added04 March 2025
Element last updated04 June 2025