Brezzi–Douglas–Durán–Fortin

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Abbreviated names	BDDF
Degrees	\(1\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Lagrange subdegree	\(\operatorname{floor}(k/3)\)
Lagrange superdegree	\(k\)
Reference cells	hexahedron
Finite dimensional space	\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(8)}_{k} \oplus \mathcal{Z}^{(9)}_{k}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(8)}_k=\operatorname{span}\left\{\nabla\times(0,0,x^{k+1}y), \nabla\times(0,xz^{k+1},0), \nabla\times(y^{k+1}z,0,0)\right\}\) \(\mathcal{Z}^{(9)}_k=\left\{\nabla\times(0,0,xy^{i+1}z^{k-i}), \nabla\times(0,x^{i+1}y^{k-i}z,0), \nabla\times(x^{k-i}yz^{i+1},0,0)\middle\|i=1,...,k\right\}\)
DOFs	On each facet: normal integral moments with a degree \(k\) Lagrange space On the interior of the reference cell: integral moments with a degree \(k-2\) vector Lagrange space
Number of DOFs	hexahedron: \((k+1)(k^2+5k+12)/2\)
Mapping	contravariant Piola
continuity	Components normal to facets are continuous
Categories	Vector-valued elements, H(div) conforming elements

Implementations

This element is implemented in Basix.UFL and Symfem .↓ Show implementation detail ↓

↑ Hide implementation detail ↑

Basix.UFL

Uses custom element code
↓ Show Basix.UFL examples ↓

Before running this example, you must install Basix.UFL:

pip3 install git+https://github.com/FEniCS/basix fenics-ufl

This element can then be created with the following lines of Python:

import basix
import basix.ufl
import numpy as np

# Create Brezzi-Douglas-Duran-Fortin degree 1 on a hexahedron
e = basix.ufl.custom_element(
    basix.CellType.hexahedron,
    (3, ),
    np.array([[0.9999999999999991, 0.0, -3.469446951953614e-17, -1.3877787807814457e-17, -2.7755575615628914e-17, 0.0, 3.122502256758253e-17, -1.3877787807814457e-17, -8.326672684688674e-17, 0.0, -6.938893903907228e-18, 0.0, -6.938893903907228e-18, 0.0, 0.0, -1.3877787807814457e-17, -2.7755575615628914e-17, 0.0, -5.204170427930421e-17, 0.0, -7.632783294297951e-17, -1.0408340855860843e-17, 0.0, -2.7755575615628914e-17, -9.367506770274758e-17, 0.0, -1.3877787807814457e-17, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.9999999999999991, 0.0, -3.469446951953614e-17, -1.3877787807814457e-17, -2.7755575615628914e-17, 0.0, 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1.3877787807814457e-17, 3.8163916471489756e-17, 0.0, -6.938893903907228e-18, -6.938893903907228e-18, 5.204170427930421e-17, -1.3877787807814457e-17, 1.3877787807814457e-17, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.3333333333333331, 0.28867513459481264, 0.07453559924999296, 3.469446951953614e-18, -3.469446951953614e-18, 0.0, -5.204170427930421e-18, -6.938893903907228e-18, -1.734723475976807e-17, -3.469446951953614e-18, -3.469446951953614e-18, 3.469446951953614e-18, 0.0, 0.0, 0.0, -1.0408340855860843e-17, 6.938893903907228e-18, 0.0, -8.673617379884035e-18, -2.0816681711721685e-17, -1.3877787807814457e-17, 0.0, 0.0, -6.938893903907228e-18, -2.0816681711721685e-17, -1.0408340855860843e-17, -1.734723475976807e-18], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.3333333333333331, 6.938893903907228e-18, -1.214306433183765e-17, 0.28867513459481264, 0.0, 3.469446951953614e-18, 0.07453559924999298, 0.0, -1.734723475976807e-17, 6.938893903907228e-18, 3.469446951953614e-18, 3.469446951953614e-18, 6.938893903907228e-18, 0.0, 0.0, -1.0408340855860843e-17, 0.0, 0.0, -1.5612511283791264e-17, 3.469446951953614e-18, -2.42861286636753e-17, -3.469446951953614e-17, 3.469446951953614e-18, 6.938893903907228e-18, -6.938893903907228e-18, 0.0, -2.42861286636753e-17, -0.49999999999999967, -0.28867513459481264, 2.0816681711721685e-17, -0.2886751345948127, -0.16666666666666655, -1.3877787807814457e-17, 1.0408340855860843e-17, 6.938893903907228e-18, 2.7755575615628914e-17, -6.938893903907228e-18, -6.938893903907228e-18, 6.938893903907228e-18, 6.938893903907228e-18, 0.0, 0.0, 2.0816681711721685e-17, 1.3877787807814457e-17, 0.0, 2.42861286636753e-17, 0.0, 4.5102810375396984e-17, 1.3877787807814457e-17, -6.938893903907228e-18, -2.7755575615628914e-17, 4.5102810375396984e-17, -2.0816681711721685e-17, 3.122502256758253e-17], [0.49999999999999967, 0.0, -1.734723475976807e-17, 0.28867513459481264, -1.3877787807814457e-17, -1.3877787807814457e-17, -3.469446951953614e-18, -6.938893903907228e-18, -4.5102810375396984e-17, 0.28867513459481264, 0.0, 1.3877787807814457e-17, 0.16666666666666655, 0.0, 0.0, 6.938893903907228e-18, -6.938893903907228e-18, 2.0816681711721685e-17, -1.734723475976807e-17, -6.938893903907228e-18, -3.469446951953614e-17, -1.3877787807814457e-17, 1.3877787807814457e-17, 2.0816681711721685e-17, -1.3877787807814457e-17, 0.0, -1.0408340855860843e-17, -0.3333333333333331, -6.938893903907228e-18, 1.214306433183765e-17, -0.28867513459481264, 0.0, -3.469446951953614e-18, -0.07453559924999298, 0.0, 1.734723475976807e-17, -6.938893903907228e-18, -3.469446951953614e-18, -3.469446951953614e-18, -6.938893903907228e-18, 0.0, 0.0, 1.0408340855860843e-17, 0.0, 0.0, 1.5612511283791264e-17, -3.469446951953614e-18, 2.42861286636753e-17, 3.469446951953614e-17, -3.469446951953614e-18, -6.938893903907228e-18, 6.938893903907228e-18, 0.0, 2.42861286636753e-17, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [-0.3333333333333331, 6.938893903907228e-18, 6.938893903907228e-18, 2.0816681711721685e-17, 0.0, 0.0, 8.673617379884035e-18, 1.0408340855860843e-17, 2.0816681711721685e-17, -0.28867513459481264, 3.469446951953614e-18, -3.469446951953614e-18, -6.938893903907228e-18, 0.0, 3.469446951953614e-18, -1.734723475976807e-17, -3.469446951953614e-18, 1.0408340855860843e-17, -0.07453559924999296, 0.0, 1.3877787807814457e-17, 6.938893903907228e-18, -3.469446951953614e-18, -3.469446951953614e-18, 2.0816681711721685e-17, 0.0, 1.0408340855860843e-17, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.49999999999999967, 0.28867513459481264, -1.734723475976807e-17, 6.938893903907228e-18, -6.938893903907228e-18, 0.0, -1.0408340855860843e-17, 6.938893903907228e-18, -4.5102810375396984e-17, 0.28867513459481264, 0.16666666666666652, 1.3877787807814457e-17, -6.938893903907228e-18, 0.0, -6.938893903907228e-18, 6.938893903907228e-18, -6.938893903907228e-18, 0.0, -1.734723475976807e-17, -1.3877787807814457e-17, -3.8163916471489756e-17, 0.0, 6.938893903907228e-18, 6.938893903907228e-18, -5.204170427930421e-17, 1.3877787807814457e-17, -1.3877787807814457e-17], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.49999999999999967, 0.28867513459481264, -2.0816681711721685e-17, 0.2886751345948127, 0.16666666666666655, 1.3877787807814457e-17, -1.0408340855860843e-17, -6.938893903907228e-18, -2.7755575615628914e-17, 6.938893903907228e-18, 6.938893903907228e-18, -6.938893903907228e-18, -6.938893903907228e-18, 0.0, 0.0, -2.0816681711721685e-17, -1.3877787807814457e-17, 0.0, -2.42861286636753e-17, 0.0, -4.5102810375396984e-17, -1.3877787807814457e-17, 6.938893903907228e-18, 2.7755575615628914e-17, -4.5102810375396984e-17, 2.0816681711721685e-17, -3.122502256758253e-17, -0.3333333333333331, -0.28867513459481264, -0.07453559924999296, -3.469446951953614e-18, 3.469446951953614e-18, 0.0, 5.204170427930421e-18, 6.938893903907228e-18, 1.734723475976807e-17, 3.469446951953614e-18, 3.469446951953614e-18, -3.469446951953614e-18, 0.0, 0.0, 0.0, 1.0408340855860843e-17, -6.938893903907228e-18, 0.0, 8.673617379884035e-18, 2.0816681711721685e-17, 1.3877787807814457e-17, 0.0, 0.0, 6.938893903907228e-18, 2.0816681711721685e-17, 1.0408340855860843e-17, 1.734723475976807e-18]], dtype=np.float64),
    [[np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64)], [np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64), np.empty((0, 3), dtype=np.float64)], [np.array([[0.21132486540518713, 0.21132486540518713, 0.0], [0.21132486540518713, 0.7886751345948129, 0.0], [0.7886751345948129, 0.21132486540518713, 0.0], [0.7886751345948129, 0.7886751345948129, 0.0]], dtype=np.float64), np.array([[0.21132486540518713, 0.0, 0.21132486540518713], [0.21132486540518713, 0.0, 0.7886751345948129], [0.7886751345948129, 0.0, 0.21132486540518713], [0.7886751345948129, 0.0, 0.7886751345948129]], dtype=np.float64), np.array([[0.0, 0.21132486540518713, 0.21132486540518713], [0.0, 0.21132486540518713, 0.7886751345948129], [0.0, 0.7886751345948129, 0.21132486540518713], [0.0, 0.7886751345948129, 0.7886751345948129]], dtype=np.float64), np.array([[1.0, 0.21132486540518713, 0.21132486540518713], [1.0, 0.21132486540518713, 0.7886751345948129], [1.0, 0.7886751345948129, 0.21132486540518713], [1.0, 0.7886751345948129, 0.7886751345948129]], dtype=np.float64), np.array([[0.21132486540518713, 1.0, 0.21132486540518713], [0.21132486540518713, 1.0, 0.7886751345948129], [0.7886751345948129, 1.0, 0.21132486540518713], [0.7886751345948129, 1.0, 0.7886751345948129]], dtype=np.float64), np.array([[0.21132486540518713, 0.21132486540518713, 1.0], [0.21132486540518713, 0.7886751345948129, 1.0], [0.7886751345948129, 0.21132486540518713, 1.0], [0.7886751345948129, 0.7886751345948129, 1.0]], dtype=np.float64)], [np.empty((0, 3), dtype=np.float64)]],
    [[np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64)], [np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64), np.empty((0, 3, 0, 1), dtype=np.float64)], [np.array([[[[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]], [[0.14433756729740643], [0.0], [0.0], [-0.14433756729740643]]], [[[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]], [[0.052831216351296784], [0.052831216351296784], [0.19716878364870322], [0.19716878364870322]]], [[[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]], [[0.052831216351296784], [0.19716878364870322], [0.052831216351296784], [0.19716878364870322]]]], dtype=np.float64), np.array([[[[0.0], [0.0], [0.0], [0.0]], [[-0.14433756729740643], [0.0], [0.0], [0.14433756729740643]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0]], [[-0.052831216351296784], [-0.052831216351296784], [-0.19716878364870322], [-0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0]], [[-0.052831216351296784], [-0.19716878364870322], [-0.052831216351296784], [-0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]]]], dtype=np.float64), np.array([[[[0.14433756729740643], [0.0], [0.0], [-0.14433756729740643]], [[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.052831216351296784], [0.052831216351296784], [0.19716878364870322], [0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.052831216351296784], [0.19716878364870322], [0.052831216351296784], [0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]]]], dtype=np.float64), np.array([[[[0.14433756729740643], [0.0], [0.0], [-0.14433756729740643]], [[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.052831216351296784], [0.052831216351296784], [0.19716878364870322], [0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.052831216351296784], [0.19716878364870322], [0.052831216351296784], [0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]]]], dtype=np.float64), np.array([[[[0.0], [0.0], [0.0], [0.0]], [[-0.14433756729740643], [0.0], [0.0], [0.14433756729740643]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0]], [[-0.052831216351296784], [-0.052831216351296784], [-0.19716878364870322], [-0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0]], [[-0.052831216351296784], [-0.19716878364870322], [-0.052831216351296784], [-0.19716878364870322]], [[0.0], [0.0], [0.0], [0.0]]]], dtype=np.float64), np.array([[[[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]], [[0.14433756729740643], [0.0], [0.0], [-0.14433756729740643]]], [[[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]], [[0.052831216351296784], [0.052831216351296784], [0.19716878364870322], [0.19716878364870322]]], [[[0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0]], [[0.052831216351296784], [0.19716878364870322], [0.052831216351296784], [0.19716878364870322]]]], dtype=np.float64)], [np.empty((0, 3, 0, 1), dtype=np.float64)]],
    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    2,
    basix.PolysetType.standard, dtype=np.float64
)

# Create Brezzi-Douglas-Duran-Fortin degree 2 on a hexahedron
e = basix.ufl.custom_element(
    basix.CellType.hexahedron,
    (3, ),
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    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    3,
    basix.PolysetType.standard, dtype=np.float64
)

Symfem

"BDDF"
↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations.

Examples

hexahedron degree 1	(click to view basis functions)
hexahedron degree 2	(click to view basis functions)

References

[1] Brezzi, Franco, Douglas, Jim, Durán, Ricardo G., and Fortin, Michel. Mixed finite elements for second order elliptic problems in three variables, Numerische Mathematik 51, 237–250, 1987. [DOI: 10.1007/BF01396752] [BibTeX]

DefElement stats

Element added	31 January 2021
Element last updated	04 June 2025