an encyclopedia of finite element definitions
Verification: full detail
Last updated: 1 January 2026| Element | Example | NDElement | Basix | Basix.UFL | FIAT | |
| discontinuous Lagrange | interval,0,equispaced | |||||
| interval,1,equispaced | ||||||
| interval,2,equispaced | ||||||
| triangle,0,equispaced | ||||||
| triangle,1,equispaced | ||||||
| triangle,2,equispaced | ||||||
| quadrilateral,0,equispaced | ||||||
| quadrilateral,1,equispaced | ||||||
| quadrilateral,2,equispaced | ||||||
| tetrahedron,0,equispaced | ||||||
| tetrahedron,1,equispaced | ||||||
| tetrahedron,2,equispaced | ||||||
| hexahedron,0,equispaced | ||||||
| hexahedron,1,equispaced | ||||||
| hexahedron,2,equispaced | ||||||
| prism,0,equispaced | ||||||
| prism,1,equispaced | ||||||
| prism,2,equispaced | ||||||
| pyramid,0,equispaced | ||||||
| pyramid,1,equispaced | ||||||
| pyramid,2,equispaced | ||||||
| Lagrange | interval,1,equispaced | |||||
| interval,2,equispaced | ||||||
| interval,3,equispaced | ||||||
| interval,1,gll | ||||||
| interval,2,gll | ||||||
| interval,3,gll | ||||||
| interval,4,gll | ||||||
| interval,1,lobatto | ||||||
| interval,2,lobatto | ||||||
| interval,3,lobatto | ||||||
| triangle,1,equispaced | ||||||
| triangle,2,equispaced | ||||||
| triangle,3,equispaced | ||||||
| quadrilateral,1,equispaced | ||||||
| quadrilateral,2,equispaced | ||||||
| quadrilateral,3,equispaced | ||||||
| quadrilateral,1,gll | ||||||
| quadrilateral,2,gll | ||||||
| quadrilateral,1,lobatto | ||||||
| quadrilateral,2,lobatto | ||||||
| quadrilateral,3,lobatto | ||||||
| tetrahedron,1,equispaced | ||||||
| tetrahedron,2,equispaced | ||||||
| hexahedron,1,equispaced | ||||||
| hexahedron,2,equispaced | ||||||
| hexahedron,1,lobatto | ||||||
| hexahedron,2,lobatto | ||||||
| prism,1,equispaced | ||||||
| prism,2,equispaced | ||||||
| pyramid,1,equispaced | ||||||
| pyramid,2,equispaced | ||||||
| Nédélec (first kind) | triangle,0,lagrange | |||||
| triangle,1,lagrange | ||||||
| triangle,0,legendre | ||||||
| triangle,1,legendre | ||||||
| quadrilateral,0,lagrange | ||||||
| quadrilateral,1,lagrange | ||||||
| quadrilateral,0,legendre | ||||||
| quadrilateral,1,legendre | ||||||
| tetrahedron,0,lagrange | ||||||
| tetrahedron,1,lagrange | ||||||
| hexahedron,0,lagrange | ||||||
| hexahedron,1,lagrange | ||||||
| prism,0,lagrange | ||||||
| prism,1,lagrange | ||||||
| Raviart–Thomas | triangle,0,lagrange | |||||
| triangle,1,lagrange | ||||||
| triangle,0,legendre | ||||||
| triangle,1,legendre | ||||||
| quadrilateral,0,lagrange | ||||||
| quadrilateral,1,lagrange | ||||||
| quadrilateral,0,legendre | ||||||
| quadrilateral,1,legendre | ||||||
| tetrahedron,0,lagrange | ||||||
| tetrahedron,1,lagrange | ||||||
| tetrahedron,0,legendre | ||||||
| tetrahedron,1,legendre | ||||||
| hexahedron,0,lagrange | ||||||
| hexahedron,1,lagrange | ||||||
| hexahedron,0,legendre | ||||||
| hexahedron,1,legendre | ||||||
| Brezzi–Douglas–Marini | triangle,1,lagrange | |||||
| triangle,2,lagrange | ||||||
| triangle,1,legendre | ||||||
| triangle,2,legendre | ||||||
| tetrahedron,1,lagrange | ||||||
| tetrahedron,2,lagrange | ||||||
| tetrahedron,1,legendre | ||||||
| tetrahedron,2,legendre | ||||||
| bubble | interval,2 | |||||
| interval,3 | ||||||
| triangle,3 | ||||||
| triangle,4 | ||||||
| Crouzeix–Raviart | triangle,1 | |||||
| quadrilateral,1 | ||||||
| tetrahedron,1 | ||||||
| hexahedron,1 | ||||||
| dPc | interval,1 | |||||
| interval,2 | ||||||
| interval,3 | ||||||
| quadrilateral,1 | ||||||
| quadrilateral,2 | ||||||
| quadrilateral,3 | ||||||
| Hellan–Herrmann–Johnson | triangle,0 | |||||
| triangle,1 | ||||||
| triangle,2 | ||||||
| tetrahedron,0 | ||||||
| tetrahedron,1 | ||||||
| tetrahedron,2 | ||||||
| Nédélec (second kind) | triangle,1,lagrange | |||||
| triangle,2,lagrange | ||||||
| triangle,1,legendre | ||||||
| triangle,2,legendre | ||||||
| tetrahedron,1,lagrange | ||||||
| tetrahedron,2,lagrange | ||||||
| tetrahedron,1,legendre | ||||||
| tetrahedron,2,legendre | ||||||
| P1-iso-P2 | interval,1 | |||||
| triangle,1 | ||||||
| quadrilateral,1 | ||||||
| Regge | triangle,1 | |||||
| triangle,2 | ||||||
| serendipity | interval,1 | |||||
| interval,2 | ||||||
| interval,3 | ||||||
| quadrilateral,1 | ||||||
| quadrilateral,2 | ||||||
| quadrilateral,3 | ||||||
| Arnold–Winther | triangle,2 | |||||
| triangle,3 | ||||||
| Bernstein | interval,1 | |||||
| interval,2 | ||||||
| interval,3 | ||||||
| triangle,1 | ||||||
| triangle,2 | ||||||
| triangle,3 | ||||||
| Hermite | interval,3 | |||||
| triangle,3 | ||||||
| tetrahedron,3 | ||||||
| Kong–Mulder–Veldhuizen | triangle,1 | |||||
| triangle,2 | ||||||
| tetrahedron,1 | ||||||
| Mardal–Tai–Winther | triangle,1 | |||||
| tetrahedron,1 | ||||||
| nonconforming Arnold–Winther | triangle,1 | |||||
| trimmed serendipity H(curl) | quadrilateral,0 | |||||
| quadrilateral,1 | ||||||
| quadrilateral,2 | ||||||
| hexahedron,0 | ||||||
| hexahedron,1 | ||||||
| hexahedron,2 | ||||||
| trimmed serendipity H(div) | quadrilateral,0 | |||||
| quadrilateral,1 | ||||||
| quadrilateral,2 | ||||||
| hexahedron,0 | ||||||
| hexahedron,1 | ||||||
| hexahedron,2 | ||||||
| Alfeld–Sorokina | triangle,2 | |||||
| Argyris | triangle,5 | |||||
| Bell | triangle,4 | |||||
| Bernardi–Raugel | triangle,1 | |||||
| tetrahedron,1 | ||||||
| tetrahedron,2 | ||||||
| Brezzi–Douglas–Durán–Fortin | hexahedron,1 | |||||
| hexahedron,2 | ||||||
| Brezzi–Douglas–Fortin–Marini | triangle,0 | |||||
| triangle,1 | ||||||
| quadrilateral,0 | ||||||
| quadrilateral,1 | ||||||
| tetrahedron,0 | ||||||
| tetrahedron,1 | ||||||
| hexahedron,0 | ||||||
| hexahedron,1 | ||||||
| bubble enriched Lagrange | triangle,1 | |||||
| triangle,2 | ||||||
| conforming Crouzeix–Raviart | triangle,1 | |||||
| triangle,2 | ||||||
| triangle,3 | ||||||
| triangle,4 | ||||||
| triangle,5 | ||||||
| Crouzeix–Falk | triangle,3 | |||||
| Gauss–Legendre | interval,1 | |||||
| interval,2 | ||||||
| quadrilateral,1 | ||||||
| quadrilateral,2 | ||||||
| Gopalakrishnan–Lederer–Schöberl | triangle,0 | |||||
| triangle,1 | ||||||
| triangle,2 | ||||||
| tetrahedron,0 | ||||||
| tetrahedron,1 | ||||||
| Guzmán–Neilan (first kind) | triangle,1 | |||||
| tetrahedron,1 | ||||||
| tetrahedron,2 | ||||||
| Guzmán–Neilan (second kind) | triangle,1 | |||||
| tetrahedron,1 | ||||||
| tetrahedron,2 | ||||||
| Hsieh–Clough–Tocher | triangle,3 | |||||
| Huang–Zhang | quadrilateral,1 | |||||
| quadrilateral,2 | ||||||
| Morley | triangle,2 | |||||
| Radau | interval,1 | |||||
| interval,2 | ||||||
| quadrilateral,1 | ||||||
| quadrilateral,2 | ||||||
| reduced Hsieh–Clough–Tocher | triangle,3 | |||||
| serendipity H(curl) | quadrilateral,1 | |||||
| quadrilateral,2 | ||||||
| hexahedron,1 | ||||||
| hexahedron,2 | ||||||
| serendipity H(div) | quadrilateral,1 | |||||
| quadrilateral,2 | ||||||
| hexahedron,1 | ||||||
| hexahedron,2 | ||||||
| Taylor | interval,1 | |||||
| interval,2 | ||||||
| interval,3 | ||||||
| triangle,1 | ||||||
| triangle,2 | ||||||
| triangle,3 | ||||||
| Tiniest tensor | quadrilateral,2 | |||||
| quadrilateral,3 | ||||||
| quadrilateral,4 | ||||||
| hexahedron,2 | ||||||
| Tiniest tensor H(curl) | quadrilateral,1 | |||||
| quadrilateral,2 | ||||||
| quadrilateral,3 | ||||||
| hexahedron,1 | ||||||
| Tiniest tensor H(div) | quadrilateral,1 | |||||
| quadrilateral,2 | ||||||
| quadrilateral,3 | ||||||
| hexahedron,1 | ||||||
| vector bubble enriched Lagrange | triangle,1 | |||||
| triangle,2 | ||||||
| vector dPc | quadrilateral,1 | |||||
| quadrilateral,2 | ||||||
| quadrilateral,3 | ||||||
| hexahedron,1 | ||||||
| hexahedron,2 | ||||||
| vector Lagrange | triangle,1 | |||||
| triangle,2 | ||||||
| tetrahedron,1 | ||||||
| tetrahedron,2 | ||||||
| vector Q | quadrilateral,1 | |||||
| quadrilateral,2 | ||||||
| hexahedron,1 | ||||||
| hexahedron,2 |
For each element in the table above, the verification test passes for an example if:
- The element's basis functions span the same space as Symfem.
- The number of DOFs associated with each sub-entity of the cell is the same as Symfem.
- The element has the same continuity between cells as Symfem.
The algorithm used to perform verification is described in detail in the DefElement paper[1].
The symbols in the table have the following meaning:
| Verification passes | |
| Verification fails | |
| Example not implemented |
You can view a summarised version of this information on the verification page.
The verification data is also available in JSON format.
Verification Github badges
| Implementation | Badge | Markdown |
| NDElement |  | [](https://defelement.org/verification/ndelement.html) |
| Basix |  | [](https://defelement.org/verification/basix.html) |
| Basix.UFL |  | [](https://defelement.org/verification/basix.ufl.html) |
| FIAT |  | [](https://defelement.org/verification/fiat.html) |
| Symfem |  | [](https://defelement.org/verification/) |
References
- [1] Scroggs, Matthew W. and Brubeck, Pablo D. and Dean, Joseph P. and Dokken, Jørgen S. and Marsden, India. DefElement: an encyclopedia of finite element definitions, submitted to Computational Science and Engineering, 2025. [DOI: 10.48550/arXiv.2506.20188]