an encyclopedia of finite element definitions
FIAT verification103 / 112
Last updated: 1 January 2026The plot above shows the number of elements passing verificiation (green line) out of the number of elements being verified (dashed black line) over time.
| Element | Example | |
| Alfeld–Sorokina | triangle,2 | |
| Argyris | triangle,5 | |
| Arnold–Winther | triangle,2 | |
| Bell | triangle,4 | |
| Bernardi–Raugel | triangle,1 | |
| tetrahedron,1 | ||
| tetrahedron,2 | ||
| Bernstein | interval,1 | |
| interval,2 | ||
| interval,3 | ||
| triangle,1 | ||
| triangle,2 | ||
| triangle,3 | ||
| Brezzi–Douglas–Marini | triangle,1,legendre | |
| triangle,2,legendre | ||
| tetrahedron,1,legendre | ||
| tetrahedron,2,legendre | ||
| bubble | interval,2 | |
| interval,3 | ||
| triangle,3 | ||
| triangle,4 | ||
| Crouzeix–Raviart | triangle,1 | |
| tetrahedron,1 | ||
| discontinuous Lagrange | interval,0,equispaced | |
| interval,1,equispaced | ||
| interval,2,equispaced | ||
| triangle,0,equispaced | ||
| triangle,1,equispaced | ||
| triangle,2,equispaced | ||
| tetrahedron,0,equispaced | ||
| tetrahedron,1,equispaced | ||
| tetrahedron,2,equispaced | ||
| dPc | interval,1 | |
| interval,2 | ||
| interval,3 | ||
| quadrilateral,1 | ||
| quadrilateral,2 | ||
| quadrilateral,3 | ||
| 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 | ||
| Hellan–Herrmann–Johnson | triangle,0 | |
| triangle,1 | ||
| triangle,2 | ||
| tetrahedron,0 | ||
| tetrahedron,1 | ||
| tetrahedron,2 | ||
| Hermite | interval,3 | |
| triangle,3 | ||
| tetrahedron,3 | ||
| Hsieh–Clough–Tocher | triangle,3 | |
| Kong–Mulder–Veldhuizen | triangle,1 | |
| triangle,2 | ||
| tetrahedron,1 | ||
| Lagrange | interval,1,equispaced | |
| interval,2,equispaced | ||
| interval,3,equispaced | ||
| triangle,1,equispaced | ||
| triangle,2,equispaced | ||
| triangle,3,equispaced | ||
| tetrahedron,1,equispaced | ||
| tetrahedron,2,equispaced | ||
| Mardal–Tai–Winther | triangle,1 | |
| Morley | triangle,2 | |
| Nédélec (first kind) | triangle,0,legendre | |
| triangle,1,legendre | ||
| Nédélec (second kind) | triangle,1,legendre | |
| triangle,2,legendre | ||
| tetrahedron,1,legendre | ||
| tetrahedron,2,legendre | ||
| nonconforming Arnold–Winther | triangle,1 | |
| P1-iso-P2 | interval,1 | |
| triangle,1 | ||
| Raviart–Thomas | triangle,0,legendre | |
| triangle,1,legendre | ||
| tetrahedron,0,legendre | ||
| tetrahedron,1,legendre | ||
| reduced Hsieh–Clough–Tocher | triangle,3 | |
| Regge | triangle,1 | |
| triangle,2 | ||
| serendipity | interval,1 | |
| interval,2 | ||
| interval,3 | ||
| quadrilateral,1 | ||
| quadrilateral,2 | ||
| quadrilateral,3 | ||
| Taylor | interval,1 | |
| interval,2 | ||
| interval,3 | ||
| triangle,1 | ||
| triangle,2 | ||
| triangle,3 | ||
| 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 |
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 |
You can information about verification of other libraries on the verification page.
Verification Github badge
| Badge | Markdown |
 | [](https://defelement.org/verification/fiat.html) |
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]