an encyclopedia of finite element definitions

# Brezzi–Douglas–Fortin–Marini

 Abbreviated names BDFM Orders $$1\leqslant k$$ Reference elements triangle, quadrilateral, tetrahedron, hexahedron Polynomial set $$\mathcal{Z}^{(10)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k-1$$ Lagrange space On the interior of the reference element: integral moments with an order $$k-2$$ vector Lagrange space Number of DOFs triangle: $$k(k+2)$$quadrilateral: $$k(k+3)$$ (A028552)tetrahedron: $$k(k+1)(k+3)/2$$hexahedron: $$k(k+1)(k+5)/2$$ Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 FIAT FIAT.BrezziDouglasFortinMarini↓ Show FIAT examples ↓ This implementation is incorrect for this element. Symfem "BDFM"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "BDFM"↓ Show (legacy) UFL examples ↓

## Examples

 triangleorder 1 (click to view basis functions) triangleorder 2 (click to view basis functions) quadrilateralorder 1 (click to view basis functions) quadrilateralorder 2 (click to view basis functions) tetrahedronorder 2 (click to view basis functions) hexahedronorder 2 (click to view basis functions)

## References

• Brezzi, Franco, Douglas, Jim, Fortin, Michel, and Marini, L. Donatella. Efficient rectangular mixed finite elements in two and three space variables, ESAIM: Mathematical Modelling and Numerical Analysis 21, 581–604, 1987. [DOI: 10.1051/m2an/1987210405811] [BibTeX]

## DefElement stats

 Element added 30 January 2021 Element last updated 16 September 2023