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| Degrees | \(0\leqslant k\)
where \(k\) is the Polynomial superdegree |
| Polynomial subdegree | \(k\) |
| Polynomial superdegree | \(k\) |
| Reference elements | triangle |
| 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 edge: integral moments of inner products of normal to edge with an degree \(k\) Lagrange space
On each face: point evaluations of tensor products with symmetric matrices whose entries are in (lagrange,k-1) |
| 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 |
This element is implemented in
FIAT ,
Symfem , and
(legacy) UFL.
↓ Show implementation detail ↓↑ Hide implementation detail ↑| 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 1
element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("triangle"), 1)
# Create Hellan-Herrmann-Johnson degree 2
element = FIAT.HellanHerrmannJohnson(FIAT.ufc_cell("triangle"), 2) |
| 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 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) |
| (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 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) |
- 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]
| Element added | 06 February 2021 |
| Element last updated | 27 September 2024 |
