Brezzi–Douglas–Fortin–Marini

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Abbreviated names	BDFM
Degrees	\(1\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Lagrange subdegree	triangle: \(k\) tetrahedron: \(k\) quadrilateral: \(\operatorname{floor}((k+1)/d)\) hexahedron: \(\operatorname{floor}((k+1)/d)\)
Lagrange superdegree	\(k+1\)
Reference cells	triangle, quadrilateral, tetrahedron, hexahedron
Finite dimensional space	\(\mathcal{Z}^{(10)}_{k+1}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{Z}^{(10)}_k=\left\{\boldsymbol{p}\in\mathcal{P}_{k}^d\middle\|\boldsymbol{p}\cdot\boldsymbol{n}\in\mathcal{P}_{k-1}\text{ on each facet}\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On 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 cell (triangle): integral moments with a degree \(k-1\) Nédélec (first kind) space On the interior of the reference cell (tetrahedron): integral moments with a degree \(k-1\) Nédélec (first kind) space On the interior of the reference cell (quadrilateral): integral moments with a degree \(k-1\) vector dPc space On the interior of the reference cell (hexahedron): integral moments with a degree \(k-1\) vector dPc space
Number of DOFs	triangle: \(k^2+5k+3\) quadrilateral: \((k+1)(k+4)\) (A028552) tetrahedron: \((k+2)(k^2+7k+4)/2\) hexahedron: \((k+1)(k+2)(k+6)/2\)
Mapping	contravariant Piola
continuity	Components normal to facets are continuous
Categories	Vector-valued elements, H(div) conforming elements

Implementations

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

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FIAT	`FIAT.BrezziDouglasFortinMarini` ↓ Show FIAT examples ↓↑ Hide FIAT examples ↑ Before running this example, you must install FIAT: pip3 install git+https://github.com/firedrakeproject/fiat.git This element can then be created with the following lines of Python: import FIAT This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: Not implemented: triangle,0; triangle,1; quadrilateral,0; quadrilateral,1; tetrahedron,0; tetrahedron,1; hexahedron,0; hexahedron,1 Note: This element uses the Lagrange superdegree as the canonical degree of this element
Symfem	`"BDFM"` ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before running this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Brezzi-Douglas-Fortin-Marini degree 0 on a triangle element = symfem.create_element("triangle", "BDFM", 0) # Create Brezzi-Douglas-Fortin-Marini degree 1 on a triangle element = symfem.create_element("triangle", "BDFM", 1) # Create Brezzi-Douglas-Fortin-Marini degree 0 on a quadrilateral element = symfem.create_element("quadrilateral", "BDFM", 0) # Create Brezzi-Douglas-Fortin-Marini degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "BDFM", 1) # Create Brezzi-Douglas-Fortin-Marini degree 0 on a tetrahedron element = symfem.create_element("tetrahedron", "BDFM", 0) # Create Brezzi-Douglas-Fortin-Marini degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "BDFM", 1) # Create Brezzi-Douglas-Fortin-Marini degree 0 on a hexahedron element = symfem.create_element("hexahedron", "BDFM", 0) # Create Brezzi-Douglas-Fortin-Marini degree 1 on a hexahedron element = symfem.create_element("hexahedron", "BDFM", 1) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"BDFM"` ↓ Show (legacy) UFL examples ↓↑ Hide (legacy) UFL examples ↑ Before running this example, you must install (legacy) UFL: pip3 install setuptools pip3 install fenics-ufl-legacy This element can then be created with the following lines of Python: import ufl_legacy # Create Brezzi-Douglas-Fortin-Marini degree 0 on a triangle element = ufl_legacy.FiniteElement("BDFM", "triangle", 1) # Create Brezzi-Douglas-Fortin-Marini degree 1 on a triangle element = ufl_legacy.FiniteElement("BDFM", "triangle", 2) # Create Brezzi-Douglas-Fortin-Marini degree 0 on a tetrahedron element = ufl_legacy.FiniteElement("BDFM", "tetrahedron", 1) # Create Brezzi-Douglas-Fortin-Marini degree 1 on a tetrahedron element = ufl_legacy.FiniteElement("BDFM", "tetrahedron", 2) Note: This element uses the Lagrange superdegree as the canonical degree of this element

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

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]
Brezzi, Franco and Fortin, Michel. Function spaces and finite element approximations, in Mixed and hybrid finite element methods (eds: Brezzi, Franco and Fortin, Michel), 89–132, 1991. [DOI: 10.1007/978-1-4612-3172-1_3] [BibTeX]

DefElement stats

Element added	30 January 2021
Element last updated	04 June 2025