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Brezzi–Douglas–Fortin–Marini

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Abbreviated namesBDFM
Degrees\(1\leqslant k\)
where \(k\) is the Polynomial subdegree
Polynomial subdegree\(k\)
Polynomial superdegree\(k+1\)
Lagrange subdegreetriangle: \(k\)
tetrahedron: \(k\)
quadrilateral: \(\operatorname{floor}((k+1)/d)\)
hexahedron: \(\operatorname{floor}((k+1)/d)\)
Lagrange superdegree\(k+1\)
Reference elementstriangle, quadrilateral, tetrahedron, hexahedron
Polynomial set\(\mathcal{Z}^{(10)}_{k+1}\)
↓ Show polynomial set definitions ↓
DOFsOn each facet (triangle): normal integral moments with a degree \(k\) Lagrange space
On each facet (tetrahedron): normal integral moments with a degree \(k\) Lagrange space
On each facet (quadrilateral): normal integral moments with a degree \(k\) dPc space
On each facet (hexahedron): normal integral moments with a degree \(k\) dPc space
On the interior of the reference element (triangle): integral moments with a degree \(k-1\) Nédélec (first kind) space
On the interior of the reference element (tetrahedron): integral moments with a degree \(k-1\) Nédélec (first kind) space
On the interior of the reference element (quadrilateral): integral moments with a degree \(k-1\) vector dPc space
On the interior of the reference element (hexahedron): integral moments with a degree \(k-1\) vector dPc space
Number of DOFstriangle: \(k^2+5k+3\)
quadrilateral: \((k+1)(k+4)\) (A028552)
tetrahedron: \((k+2)(k^2+7k+4)/2\)
hexahedron: \((k+1)(k+2)(k+6)/2\)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

This element is implemented in FIAT , Symfem , and (legacy) UFL.↓ Show implementation detail ↓

Examples

triangle
degree 0

(click to view basis functions)
triangle
degree 1

(click to view basis functions)
quadrilateral
degree 0

(click to view basis functions)
quadrilateral
degree 1

(click to view basis functions)
tetrahedron
degree 0

(click to view basis functions)
tetrahedron
degree 1

(click to view basis functions)
hexahedron
degree 0

(click to view basis functions)
hexahedron
degree 1

(click to view basis functions)

References

DefElement stats

Element added30 January 2021
Element last updated31 March 2025