an encyclopedia of finite element definitions

# Hellan–Herrmann–Johnson

 Orders $$0\leqslant k$$ Reference elements triangle Polynomial set $$\mathcal{Z}^{(2)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each edge: integral moments of inner products of normal to edge with an order $$k$$ 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

## Implementations

 FIAT FIAT.HellanHerrmannJohnson↓ Show FIAT examples ↓ This implementation is correct for all the examples below. Symfem "HHJ"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "HHJ"↓ Show (legacy) UFL examples ↓

## Examples

 triangleorder 1 (click to view basis functions) triangleorder 2 (click to view basis functions)
• 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]

## DefElement stats

 Element added 06 February 2021 Element last updated 16 September 2023