Brezzi–Douglas–Marini

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De Rham complex families	\(\left[S_{1,k}^\unicode{0x25FA}\right]_{d-1}\) or \(\mathcal{P}_{k}\Lambda^{d-1}(\Delta_d)\)
Abbreviated names	BDM
Variants	Legendre: Integral moments are taken against orthonormal polynomials Lagrange: Integral moments are taken against Lagrange basis functions
Degrees	\(1\leqslant k\) where \(k\) is the Polynomial superdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k\)
Reference elements	triangle, tetrahedron
Polynomial set	\(\mathcal{P}_{k}^d\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On each facet: normal integral moments with an degree \(k\) Lagrange space On the interior of the reference element: integral moments with an degree \(k-1\) Nédélec (first kind) space
Number of DOFs	triangle: \((k+1)(k+2)\) (A002378) tetrahedron: \((k+1)(k+2)(k+3)/2\) (A027480)
Mapping	contravariant Piola
continuity	Components normal to facets are continuous
Categories	Vector-valued elements, H(div) conforming elements

Implementations

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

↑ Hide implementation detail ↑

Basix	`basix.ElementFamily.BDM` ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before running this example, you must install Basix: pip3 install fenics-basix This element can then be created with the following lines of Python: import basix # Create Brezzi-Douglas-Marini (Lagrange variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.triangle, 1) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.triangle, 2) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 1) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 2) # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.legendre) # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.legendre) This implementation is correct for all the examples below.
Basix.UFL	`basix.ElementFamily.BDM` ↓ Show Basix.UFL examples ↓↑ Hide Basix.UFL examples ↑ Before running this example, you must install Basix.UFL: pip3 install fenics-basix fenics-ufl This element can then be created with the following lines of Python: import basix import basix.ufl # Create Brezzi-Douglas-Marini (Lagrange variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.triangle, 1) # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.triangle, 2) # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 1) # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.BDM, basix.CellType.tetrahedron, 2) This implementation is correct for all the examples below.
FIAT	`FIAT.BrezziDouglasMarini(..., variant="integral")` ↓ 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 # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 element = FIAT.BrezziDouglasMarini(FIAT.ufc_cell("triangle"), 1, variant="integral") # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 element = FIAT.BrezziDouglasMarini(FIAT.ufc_cell("triangle"), 2, variant="integral") # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 element = FIAT.BrezziDouglasMarini(FIAT.ufc_cell("tetrahedron"), 1, variant="integral") # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 element = FIAT.BrezziDouglasMarini(FIAT.ufc_cell("tetrahedron"), 2, variant="integral") This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: triangle,1,legendre; triangle,2,legendre; tetrahedron,1,legendre; tetrahedron,2,legendre Not implemented: triangle,1,lagrange; triangle,2,lagrange; tetrahedron,1,lagrange; tetrahedron,2,lagrange
Symfem	`"N2div", variant="legendre"` (Legendre) `"N2div"` (Lagrange) ↓ 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-Marini (Lagrange variant) degree 1 on a triangle element = symfem.create_element("triangle", "N2div", 1) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a triangle element = symfem.create_element("triangle", "N2div", 2) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "N2div", 1) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "N2div", 2) # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 on a triangle element = symfem.create_element("triangle", "N2div", 1, variant="legendre") # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 on a triangle element = symfem.create_element("triangle", "N2div", 2, variant="legendre") # Create Brezzi-Douglas-Marini (Legendre variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "N2div", 1, variant="legendre") # Create Brezzi-Douglas-Marini (Legendre variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "N2div", 2, variant="legendre") This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"BDM"` ↓ 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-Marini (Lagrange variant) degree 1 on a triangle element = ufl_legacy.FiniteElement("BDM", "triangle", 1) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a triangle element = ufl_legacy.FiniteElement("BDM", "triangle", 2) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 1 on a tetrahedron element = ufl_legacy.FiniteElement("BDM", "tetrahedron", 1) # Create Brezzi-Douglas-Marini (Lagrange variant) degree 2 on a tetrahedron element = ufl_legacy.FiniteElement("BDM", "tetrahedron", 2)

Examples

triangle degree 1 Lagrange variant	(click to view basis functions)
triangle degree 2 Lagrange variant	(click to view basis functions)
tetrahedron degree 1 Lagrange variant	(click to view basis functions)
tetrahedron degree 2 Lagrange variant	(click to view basis functions)
triangle degree 1 Legendre variant	(click to view basis functions)
triangle degree 2 Legendre variant	(click to view basis functions)
tetrahedron degree 1 Legendre variant	(click to view basis functions)
tetrahedron degree 2 Legendre variant	(click to view basis functions)

References

Brezzi, Franco, Douglas, Jim, and Marini, L. Donatella. Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik 47, 217–235, 1985. [DOI: 10.1007/BF01389710] [BibTeX]
Arnold, Douglas N. and Logg, Anders. Periodic table of the finite elements, SIAM News 47, 2014. [www.siam.org/publications/siam-news/issues/volume-47-number-09-november-2014/] [BibTeX]
Cockburn, Bernardo and Fu, Guosheng. A systematic construction of finite element commuting exact sequences, SIAM journal on numerical analysis 55, 1650–1688, 2017. [DOI: 10.1137/16M1073352] [BibTeX]

DefElement stats

Element added	30 December 2020
Element last updated	27 September 2024