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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 cells	triangle, tetrahedron
Polynomial set	\(\mathcal{Z}^{(2)}_{k}\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{Z}^{(2)}_k=\left\{\mathbf{M}\in\mathcal{P}_{k}^{d\times d}\middle\|\mathbf{M}^t=\mathbf{M}\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On 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 DOFs	triangle: \(3(k+1)(k+2)/2\) (A045943)
Mapping	double contravariant Piola
continuity	Inner products with normals to facets are continuous
Categories	Matrix-valued elements

Implementations

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

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Basix	`basix.ElementFamily.HHJ` ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before running this example, you must install Basix: pip3 install git+https://github.com/FEniCS/basix This element can then be created with the following lines of Python: import basix # Create Hellan-Herrmann-Johnson degree 0 on a triangle element = basix.create_element(basix.ElementFamily.HHJ, basix.CellType.triangle, 0) # Create Hellan-Herrmann-Johnson degree 1 on a triangle element = basix.create_element(basix.ElementFamily.HHJ, basix.CellType.triangle, 1) # Create Hellan-Herrmann-Johnson degree 2 on a triangle element = basix.create_element(basix.ElementFamily.HHJ, basix.CellType.triangle, 2) # Create Hellan-Herrmann-Johnson degree 0 on a tetrahedron element = basix.create_element(basix.ElementFamily.HHJ, basix.CellType.tetrahedron, 0) # Create Hellan-Herrmann-Johnson degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.HHJ, basix.CellType.tetrahedron, 1) # Create Hellan-Herrmann-Johnson degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.HHJ, basix.CellType.tetrahedron, 2) This implementation is correct for all the examples below.
Basix.UFL	`basix.ElementFamily.HHJ` ↓ Show Basix.UFL examples ↓↑ Hide Basix.UFL examples ↑ Before running this example, you must install Basix.UFL: pip3 install git+https://github.com/FEniCS/basix fenics-ufl This element can then be created with the following lines of Python: import basix import basix.ufl # Create Hellan-Herrmann-Johnson degree 0 on a triangle element = basix.ufl.element(basix.ElementFamily.HHJ, basix.CellType.triangle, 0) # Create Hellan-Herrmann-Johnson degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.HHJ, basix.CellType.triangle, 1) # Create Hellan-Herrmann-Johnson degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.HHJ, basix.CellType.triangle, 2) # Create Hellan-Herrmann-Johnson degree 0 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.HHJ, basix.CellType.tetrahedron, 0) # Create Hellan-Herrmann-Johnson degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.HHJ, basix.CellType.tetrahedron, 1) # Create Hellan-Herrmann-Johnson degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.HHJ, basix.CellType.tetrahedron, 2) This implementation is correct for all the examples below.
FIAT	`FIAT.HellanHerrmannJohnson` ↓ Show FIAT examples ↓↑ Hide FIAT examples ↑ Before running this example, you must install FIAT: pip3 install git+https://github.com/firedrakeproject/fiat.git This element can then be created with the following lines of Python: import FIAT # Create Hellan-Herrmann-Johnson degree 0 element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("triangle"), 0) # Create Hellan-Herrmann-Johnson degree 1 element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("triangle"), 1) # Create Hellan-Herrmann-Johnson degree 2 element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("triangle"), 2) # Create Hellan-Herrmann-Johnson degree 0 element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("tetrahedron"), 0) # Create Hellan-Herrmann-Johnson degree 1 element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("tetrahedron"), 1) # Create Hellan-Herrmann-Johnson degree 2 element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("tetrahedron"), 2) This implementation is correct for all the examples below.
Symfem	`"HHJ"` ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before running this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Hellan-Herrmann-Johnson degree 0 on a triangle element = symfem.create_element("triangle", "HHJ", 0) # Create Hellan-Herrmann-Johnson degree 1 on a triangle element = symfem.create_element("triangle", "HHJ", 1) # Create Hellan-Herrmann-Johnson degree 2 on a triangle element = symfem.create_element("triangle", "HHJ", 2) # Create Hellan-Herrmann-Johnson degree 0 on a tetrahedron element = symfem.create_element("tetrahedron", "HHJ", 0) # Create Hellan-Herrmann-Johnson degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "HHJ", 1) # Create Hellan-Herrmann-Johnson degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "HHJ", 2) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"HHJ"` ↓ Show (legacy) UFL examples ↓↑ Hide (legacy) UFL examples ↑ Before running this example, you must install (legacy) UFL: pip3 install setuptools pip3 install fenics-ufl-legacy This element can then be created with the following lines of Python: import ufl_legacy # Create Hellan-Herrmann-Johnson degree 0 on a triangle element = ufl_legacy.FiniteElement("HHJ", "triangle", 0) # Create Hellan-Herrmann-Johnson degree 1 on a triangle element = ufl_legacy.FiniteElement("HHJ", "triangle", 1) # Create Hellan-Herrmann-Johnson degree 2 on a triangle element = ufl_legacy.FiniteElement("HHJ", "triangle", 2)

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

Arnold, Douglas N. and Walker, Shawn W. The Hellan–Herrmann–Johnson method with curved elements, SIAM Journal on Numberical Analysis 58(5), 2829–2855, 2020. [DOI: 10.1137/19M1288723] [BibTeX]
Pechstein, Astrid S. and Schöberl, Joachim. An analysis of the TDNNS method using natural norms, Numerische Mathematik 139, 92–120, 2018. [DOI: 10.1007/s00211-017-0933-3] [BibTeX]

DefElement stats

Element added	04 March 2025
Element last updated	04 June 2025