Brezzi–Douglas–Fortin–Marini

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Abbreviated names	BDFM
Degrees	$1 ⩽ k$ where $k$ is the Polynomial subdegree
Polynomial subdegree	$k$
Polynomial superdegree	$k + 1$
Lagrange subdegree	triangle: $k$ tetrahedron: $k$ quadrilateral: $floor ((k + 1) / d)$ hexahedron: $floor ((k + 1) / d)$
Lagrange superdegree	$k + 1$
Reference elements	triangle, quadrilateral, tetrahedron, hexahedron
Polynomial set	$Z_{k + 1}^{(10)}$ ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ $Z_{k}^{(10)} = {p \in P_{k}^{d} \| p \cdot n \in P_{k - 1} on each facet}$ $P_{k} = span {\prod_{i = 1}^{d} x_{i}^{p_{i}} \| \sum_{i = 1}^{d} p_{i} ⩽ k}$
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 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 DOFs	triangle: $k^{2} + 5 k + 3$ quadrilateral: $(k + 1) (k + 4)$ (A028552) tetrahedron: $(k + 2) (k^{2} + 7 k + 4) / 2$ 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 ↓

↑ Hide implementation detail ↑

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
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)

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), 1991. [DOI: 10.1007/978-1-4612-3172-1_3] [BibTeX]

DefElement stats

Element added	30 January 2021
Element last updated	31 March 2025