Serendipity H(div)

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Alternative names	Brezzi–Douglas–Marini cubical H(div) (quadrilateral), Arnold–Awanou H(div) (hexahedron)
De Rham complex families	\(\left[S_{1,k}^\square\right]_{d-1}\) or \(\mathcal{S}_{k}\Lambda^{d-1}(\square_d)\)
Abbreviated names	BDMcf (quadrilateral), AAf (hexahedron)
Degrees	\(1\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Lagrange subdegree	\(\operatorname{floor}(k/d)\)
Lagrange superdegree	\(k+1\)
Reference cells	quadrilateral, hexahedron
Finite dimensional space	\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(28)}_{k}\) (quadrilateral) \(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(29)}_{k}\) (hexahedron) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(28)}_k=\operatorname{span}\left\{\left(\begin{array}{c}x^{k+1}\\(k+1)x^ky\end{array}\right),\left(\begin{array}{c}(k+1)xy^k\\y^{k+1}\end{array}\right)\right\}\) \(\mathcal{Z}^{(29)}_k=\left\{\nabla\times\boldsymbol{p}\middle\|\boldsymbol{p}\in\mathcal{A}_{k}\right\}\) \(\mathcal{A}_k=\operatorname{span}\bigcup_{p\in\mathcal{P}_k(\mathbb{R}^{d-1})}\left\{\left(\begin{array}{c}0\\xzp(y,z)\\-xyp(y,z)\end{array}\right),\left(\begin{array}{c}yzp(x,z)\\0\\-xyp(x,z)\end{array}\right),\left(\begin{array}{c}yzp(x,y)\\-xzp(x,y)\\0\end{array}\right)\right\}\)
DOFs	On each facet: normal integral moments with a degree \(k\) dPc space On the interior of the reference cell: integral moments with a degree \(k-2\) vector dPc space
Number of DOFs	quadrilateral: \(k^2+3k+4\) (A014206) 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 serendipity H(div) degree 1 on a quadrilateral
e = basix.ufl.custom_element(
    basix.CellType.quadrilateral,
    (2, ),
    np.array([[0.9999999999999996, 0.0, 0.0, -2.7755575615628914e-17, 0.0, 0.0, 0.0, 0.0, -8.326672684688674e-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, 0.9999999999999996, 0.0, 0.0, -2.7755575615628914e-17, 0.0, 0.0, 0.0, 0.0, -8.326672684688674e-17], [0.49999999999999983, 0.0, -1.3877787807814457e-17, 0.28867513459481275, 0.0, -2.7755575615628914e-17, 0.0, 1.3877787807814457e-17, -4.163336342344337e-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, 0.49999999999999983, 0.0, -1.3877787807814457e-17, 0.28867513459481275, 0.0, -2.7755575615628914e-17, 0.0, 1.3877787807814457e-17, -4.163336342344337e-17], [0.4999999999999998, 0.28867513459481275, 0.0, 0.0, 0.0, 0.0, 1.3877787807814457e-17, -4.163336342344337e-17, -4.163336342344337e-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, 0.4999999999999998, 0.28867513459481275, 0.0, 0.0, 0.0, 0.0, 1.3877787807814457e-17, -4.163336342344337e-17, -4.163336342344337e-17], [0.33333333333333315, 0.0, -1.3877787807814457e-17, 0.2886751345948127, 0.0, -2.7755575615628914e-17, 0.07453559924999298, 0.0, -2.0816681711721685e-17, 0.4999999999999998, 0.2886751345948127, 0.0, 0.28867513459481264, 0.16666666666666657, 0.0, 1.3877787807814457e-17, -5.551115123125783e-17, -2.7755575615628914e-17], [0.4999999999999998, 0.2886751345948127, 0.0, 0.28867513459481264, 0.16666666666666657, 0.0, 1.3877787807814457e-17, -5.551115123125783e-17, -2.7755575615628914e-17, 0.33333333333333315, 0.2886751345948127, 0.07453559924999298, 0.0, 0.0, 0.0, -1.3877787807814457e-17, -4.163336342344337e-17, -1.3877787807814457e-17]], dtype=np.float64),
    [[np.empty((0, 2), dtype=np.float64), np.empty((0, 2), dtype=np.float64), np.empty((0, 2), dtype=np.float64), np.empty((0, 2), dtype=np.float64)], [np.array([[0.21132486540518713, 0.0], [0.7886751345948129, 0.0]], dtype=np.float64), np.array([[0.0, 0.21132486540518713], [0.0, 0.7886751345948129]], dtype=np.float64), np.array([[1.0, 0.21132486540518713], [1.0, 0.7886751345948129]], dtype=np.float64), np.array([[0.21132486540518713, 1.0], [0.7886751345948129, 1.0]], dtype=np.float64)], [np.empty((0, 2), dtype=np.float64)], []],
    [[np.empty((0, 2, 0, 1), dtype=np.float64), np.empty((0, 2, 0, 1), dtype=np.float64), np.empty((0, 2, 0, 1), dtype=np.float64), np.empty((0, 2, 0, 1), dtype=np.float64)], [np.array([[[[0.0], [0.0]], [[0.39433756729740643], [0.10566243270259357]]], [[[0.0], [0.0]], [[0.10566243270259357], [0.39433756729740643]]]], dtype=np.float64), np.array([[[[-0.39433756729740643], [-0.10566243270259357]], [[0.0], [0.0]]], [[[-0.10566243270259357], [-0.39433756729740643]], [[0.0], [0.0]]]], dtype=np.float64), np.array([[[[-0.39433756729740643], [-0.10566243270259357]], [[0.0], [0.0]]], [[[-0.10566243270259357], [-0.39433756729740643]], [[0.0], [0.0]]]], dtype=np.float64), np.array([[[[0.0], [0.0]], [[0.39433756729740643], [0.10566243270259357]]], [[[0.0], [0.0]], [[0.10566243270259357], [0.39433756729740643]]]], dtype=np.float64)], [np.empty((0, 2, 0, 1), dtype=np.float64)], []],
    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    2,
    basix.PolysetType.standard, dtype=np.float64
)

# Create serendipity H(div) degree 2 on a quadrilateral
e = basix.ufl.custom_element(
    basix.CellType.quadrilateral,
    (2, ),
    np.array([[0.9999999999999996, -8.326672684688674e-17, -3.608224830031759e-16, -1.942890293094024e-16, -9.71445146547012e-17, 0.0, 6.938893903907228e-18, 2.0816681711721685e-17, -3.885780586188048e-16, 0.0, 6.938893903907228e-18, -4.163336342344337e-17, -2.220446049250313e-16, 8.326672684688674e-17, -4.163336342344337e-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, 0.9999999999999996, -8.326672684688674e-17, -3.608224830031759e-16, -1.942890293094024e-16, -9.71445146547012e-17, 0.0, 6.938893903907228e-18, 2.0816681711721685e-17, -3.885780586188048e-16, 0.0, 6.938893903907228e-18, -4.163336342344337e-17, -2.220446049250313e-16, 8.326672684688674e-17, -4.163336342344337e-17, 0.0], [0.49999999999999967, -2.7755575615628914e-17, -1.942890293094024e-16, -9.020562075079397e-17, 0.28867513459481264, -2.7755575615628914e-17, -1.1796119636642288e-16, -6.591949208711867e-17, -2.7755575615628914e-16, 0.0, -1.3877787807814457e-17, -2.42861286636753e-17, -3.3306690738754696e-16, 1.3877787807814457e-17, -4.85722573273506e-17, -6.938893903907228e-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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.49999999999999967, -2.7755575615628914e-17, -1.942890293094024e-16, -9.020562075079397e-17, 0.28867513459481264, -2.7755575615628914e-17, -1.1796119636642288e-16, -6.591949208711867e-17, -2.7755575615628914e-16, 0.0, -1.3877787807814457e-17, -2.42861286636753e-17, -3.3306690738754696e-16, 1.3877787807814457e-17, -4.85722573273506e-17, -6.938893903907228e-18], [0.33333333333333315, -2.7755575615628914e-17, -1.1796119636642288e-16, -7.632783294297951e-17, 0.28867513459481264, -2.7755575615628914e-17, -1.1449174941446927e-16, -6.245004513516506e-17, 0.07453559924999274, 1.3877787807814457e-17, -5.204170427930421e-17, -6.938893903907228e-18, -3.3306690738754696e-16, 1.3877787807814457e-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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.33333333333333315, -2.7755575615628914e-17, -1.1796119636642288e-16, -7.632783294297951e-17, 0.28867513459481264, -2.7755575615628914e-17, -1.1449174941446927e-16, -6.245004513516506e-17, 0.07453559924999274, 1.3877787807814457e-17, -5.204170427930421e-17, -6.938893903907228e-18, -3.3306690738754696e-16, 1.3877787807814457e-17, -1.3877787807814457e-17, -1.3877787807814457e-17], [0.4999999999999997, 0.28867513459481264, -2.706168622523819e-16, -3.3306690738754696e-16, -4.163336342344337e-17, -2.7755575615628914e-17, -8.673617379884035e-19, 7.806255641895632e-18, -1.95590071916385e-16, -9.627715291671279e-17, 2.6020852139652106e-18, -3.2959746043559335e-17, -8.326672684688674e-17, -2.7755575615628914e-17, -3.8163916471489756e-17, -4.163336342344337e-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, 0.4999999999999997, 0.28867513459481264, -2.706168622523819e-16, -3.3306690738754696e-16, -4.163336342344337e-17, -2.7755575615628914e-17, -8.673617379884035e-19, 7.806255641895632e-18, -1.95590071916385e-16, -9.627715291671279e-17, 2.6020852139652106e-18, -3.2959746043559335e-17, -8.326672684688674e-17, -2.7755575615628914e-17, -3.8163916471489756e-17, -4.163336342344337e-17], [0.24999999999999983, 0.14433756729740632, -1.3704315460216776e-16, -1.5265566588595902e-16, 0.1443375672974063, 0.0833333333333332, -7.45931094670027e-17, -7.112366251504909e-17, -1.249000902703301e-16, -6.245004513516506e-17, 1.734723475976807e-18, -1.734723475976807e-18, -1.6653345369377348e-16, -5.551115123125783e-17, -2.42861286636753e-17, -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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.24999999999999983, 0.14433756729740632, -1.3704315460216776e-16, -1.5265566588595902e-16, 0.1443375672974063, 0.0833333333333332, -7.45931094670027e-17, -7.112366251504909e-17, -1.249000902703301e-16, -6.245004513516506e-17, 1.734723475976807e-18, -1.734723475976807e-18, -1.6653345369377348e-16, -5.551115123125783e-17, -2.42861286636753e-17, -2.42861286636753e-17], [0.33333333333333315, 0.28867513459481264, 0.07453559924999274, -3.434752482434078e-16, -2.7755575615628914e-17, -3.122502256758253e-17, 1.3877787807814457e-17, 8.673617379884035e-19, -1.2354483755472323e-16, -1.1053441148489718e-16, -2.7321894746634712e-17, -1.0272815584300155e-17, -5.551115123125783e-17, -2.7755575615628914e-17, -3.469446951953614e-17, -4.163336342344337e-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, 0.33333333333333315, 0.28867513459481264, 0.07453559924999274, -3.434752482434078e-16, -2.7755575615628914e-17, -3.122502256758253e-17, 1.3877787807814457e-17, 8.673617379884035e-19, -1.2354483755472323e-16, -1.1053441148489718e-16, -2.7321894746634712e-17, -1.0272815584300155e-17, -5.551115123125783e-17, -2.7755575615628914e-17, -3.469446951953614e-17, -4.163336342344337e-17], [0.24999999999999978, -6.938893903907228e-18, -1.0061396160665481e-16, -4.5102810375396984e-17, 0.2598076211353314, -1.734723475976807e-17, -1.1622647289044608e-16, -5.551115123125783e-17, 0.11180339887498919, -6.938893903907228e-18, -6.591949208711867e-17, -2.949029909160572e-17, 0.01889822365046101, 1.3877787807814457e-17, -3.469446951953614e-17, -1.0408340855860843e-17, 0.4999999999999997, 0.28867513459481264, -2.636779683484747e-16, -3.2612801348363973e-16, 0.43301270189221897, 0.24999999999999964, -2.0816681711721685e-16, -2.42861286636753e-16, 0.11180339887498908, 0.06454972243679004, -6.938893903907228e-17, -8.326672684688674e-17, -5.273559366969494e-16, -2.498001805406602e-16, -5.898059818321144e-17, -6.938893903907228e-17], [0.49999999999999967, 0.43301270189221897, 0.11180339887498907, -5.065392549852277e-16, 0.28867513459481253, 0.24999999999999967, 0.06454972243679, -2.723515857283587e-16, -2.498001805406602e-16, -1.5265566588595902e-16, -4.163336342344337e-17, -3.469446951953614e-17, -3.0531133177191805e-16, -1.942890293094024e-16, -1.1796119636642288e-16, -4.85722573273506e-17, 0.24999999999999983, 0.2598076211353313, 0.11180339887498922, 0.018898223650461014, -2.0816681711721685e-17, -2.7755575615628914e-17, -1.1275702593849246e-17, 3.0357660829594124e-18, -9.890126098730662e-17, -1.0231141563295243e-16, -3.94920641327845e-17, -1.473498515043581e-17, -4.163336342344337e-17, -2.7755575615628914e-17, -3.469446951953614e-17, -1.3877787807814457e-17]], dtype=np.float64),
    [[np.empty((0, 2), dtype=np.float64), np.empty((0, 2), dtype=np.float64), np.empty((0, 2), dtype=np.float64), np.empty((0, 2), dtype=np.float64)], [np.array([[0.1127016653792583, 0.0], [0.5, 0.0], [0.8872983346207417, 0.0]], dtype=np.float64), np.array([[0.0, 0.1127016653792583], [0.0, 0.5], [0.0, 0.8872983346207417]], dtype=np.float64), np.array([[1.0, 0.1127016653792583], [1.0, 0.5], [1.0, 0.8872983346207417]], dtype=np.float64), np.array([[0.1127016653792583, 1.0], [0.5, 1.0], [0.8872983346207417, 1.0]], dtype=np.float64)], [np.array([[0.21132486540518713, 0.21132486540518713], [0.21132486540518713, 0.7886751345948129], [0.7886751345948129, 0.21132486540518713], [0.7886751345948129, 0.7886751345948129]], dtype=np.float64)], []],
    [[np.empty((0, 2, 0, 1), dtype=np.float64), np.empty((0, 2, 0, 1), dtype=np.float64), np.empty((0, 2, 0, 1), dtype=np.float64), np.empty((0, 2, 0, 1), dtype=np.float64)], [np.array([[[[0.0], [0.0], [0.0]], [[0.1909162040613171], [0.0], [-0.024249537394650453]]], [[[0.0], [0.0], [0.0]], [[-0.02424953739465046], [0.0], [0.1909162040613171]]], [[[0.0], [0.0], [0.0]], [[0.11111111111111106], [0.4444444444444444], [0.11111111111111105]]]], dtype=np.float64), np.array([[[[-0.1909162040613171], [0.0], [0.024249537394650515]], [[0.0], [0.0], [0.0]]], [[[0.02424953739465046], [0.0], [-0.1909162040613171]], [[0.0], [0.0], [0.0]]], [[[-0.11111111111111106], [-0.4444444444444444], [-0.11111111111111105]], [[0.0], [0.0], [0.0]]]], dtype=np.float64), np.array([[[[-0.1909162040613171], [0.0], [0.024249537394650515]], [[0.0], [0.0], [0.0]]], [[[0.02424953739465046], [0.0], [-0.1909162040613171]], [[0.0], [0.0], [0.0]]], [[[-0.11111111111111106], [-0.4444444444444444], [-0.11111111111111105]], [[0.0], [0.0], [0.0]]]], dtype=np.float64), np.array([[[[0.0], [0.0], [0.0]], [[0.1909162040613171], [0.0], [-0.024249537394650453]]], [[[0.0], [0.0], [0.0]], [[-0.02424953739465046], [0.0], [0.1909162040613171]]], [[[0.0], [0.0], [0.0]], [[0.11111111111111106], [0.4444444444444444], [0.11111111111111105]]]], dtype=np.float64)], [np.array([[[[0.25], [0.25], [0.25], [0.25]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0]], [[0.25], [0.25], [0.25], [0.25]]]], dtype=np.float64)], []],
    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    3,
    basix.PolysetType.standard, dtype=np.float64
)

# Create serendipity H(div) degree 1 on a hexahedron
e = basix.ufl.custom_element(
    basix.CellType.hexahedron,
    (3, ),
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-6.938893903907228e-18, 6.938893903907228e-18, 1.0408340855860843e-17, -6.938893903907228e-18, 0.0, -5.204170427930421e-18, 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, -0.7499999999999994, -0.4330127018922191, 2.7755575615628914e-17, -0.43301270189221897, -0.24999999999999983, -4.163336342344337e-17, 6.938893903907228e-17, 0.0, 5.551115123125783e-17, -1.3877787807814457e-17, 0.0, 0.0, -2.7755575615628914e-17, 0.0, 0.0, -2.7755575615628914e-17, 0.0, -6.938893903907228e-18, 4.163336342344337e-17, 5.551115123125783e-17, 4.163336342344337e-17, 3.469446951953614e-17, 0.0, 0.0, 4.163336342344337e-17, 2.0816681711721685e-17, -1.3877787807814457e-17], [-0.7499999999999994, 0.0, 3.469446951953614e-17, -0.43301270189221897, 0.0, -2.0816681711721685e-17, 3.469446951953614e-17, 0.0, 5.551115123125783e-17, -0.43301270189221897, 0.0, -1.3877787807814457e-17, -0.24999999999999983, 0.0, 0.0, -2.7755575615628914e-17, 0.0, 2.0816681711721685e-17, 3.469446951953614e-17, 0.0, 5.551115123125783e-17, -1.3877787807814457e-17, -1.3877787807814457e-17, 2.7755575615628914e-17, 6.938893903907228e-17, 0.0, -4.85722573273506e-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, 0.24999999999999983, 0.14433756729740632, -1.0408340855860843e-17, 0.14433756729740635, 0.08333333333333327, 6.938893903907228e-18, -5.204170427930421e-18, -3.469446951953614e-18, -1.3877787807814457e-17, 3.469446951953614e-18, 3.469446951953614e-18, -3.469446951953614e-18, -3.469446951953614e-18, 0.0, 0.0, -1.0408340855860843e-17, -6.938893903907228e-18, 0.0, -1.214306433183765e-17, 0.0, -2.2551405187698492e-17, -6.938893903907228e-18, 3.469446951953614e-18, 1.3877787807814457e-17, -2.2551405187698492e-17, 1.0408340855860843e-17, -1.5612511283791264e-17], [-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, 0.7499999999999994, 0.0, -3.469446951953614e-17, 0.43301270189221897, 0.0, 2.0816681711721685e-17, -3.469446951953614e-17, 0.0, -5.551115123125783e-17, 0.43301270189221897, 0.0, 1.3877787807814457e-17, 0.24999999999999983, 0.0, 0.0, 2.7755575615628914e-17, 0.0, -2.0816681711721685e-17, -3.469446951953614e-17, 0.0, -5.551115123125783e-17, 1.3877787807814457e-17, 1.3877787807814457e-17, -2.7755575615628914e-17, -6.938893903907228e-17, 0.0, 4.85722573273506e-17, -0.24999999999999983, -0.14433756729740632, 8.673617379884035e-18, -3.469446951953614e-18, 3.469446951953614e-18, 0.0, 5.204170427930421e-18, -3.469446951953614e-18, 2.2551405187698492e-17, -0.14433756729740632, -0.08333333333333326, -6.938893903907228e-18, 3.469446951953614e-18, 0.0, 3.469446951953614e-18, -3.469446951953614e-18, 3.469446951953614e-18, 0.0, 8.673617379884035e-18, 6.938893903907228e-18, 1.9081958235744878e-17, 0.0, -3.469446951953614e-18, -3.469446951953614e-18, 2.6020852139652106e-17, -6.938893903907228e-18, 6.938893903907228e-18], [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, 0.24999999999999983, 0.0, -8.673617379884035e-18, 0.14433756729740632, -6.938893903907228e-18, -6.938893903907228e-18, -1.734723475976807e-18, -3.469446951953614e-18, -2.2551405187698492e-17, 0.14433756729740632, 0.0, 6.938893903907228e-18, 0.08333333333333327, 0.0, 0.0, 3.469446951953614e-18, -3.469446951953614e-18, 1.0408340855860843e-17, -8.673617379884035e-18, -3.469446951953614e-18, -1.734723475976807e-17, -6.938893903907228e-18, 6.938893903907228e-18, 1.0408340855860843e-17, -6.938893903907228e-18, 0.0, -5.204170427930421e-18, -0.7499999999999994, -0.43301270189221897, 2.7755575615628914e-17, 4.163336342344337e-17, 1.3877787807814457e-17, 0.0, -1.3877787807814457e-17, 5.551115123125783e-17, 4.163336342344337e-17, -0.43301270189221897, -0.24999999999999983, -4.163336342344337e-17, 0.0, 0.0, 1.3877787807814457e-17, -2.7755575615628914e-17, 2.7755575615628914e-17, 2.7755575615628914e-17, 1.3877787807814457e-17, 2.7755575615628914e-17, 5.551115123125783e-17, 6.938893903907228e-18, 0.0, -1.3877787807814457e-17, 4.163336342344337e-17, 2.7755575615628914e-17, -3.469446951953614e-17]], 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 serendipity H(div) degree 2 on a hexahedron
e = basix.ufl.custom_element(
    basix.CellType.hexahedron,
    (3, ),
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[0.24299331315454972]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[-0.006735982609625124], [0.0], [0.05303227890592139], [-0.010777572175400204], [0.0], [0.08485164624947425], [-0.006735982609625124], [0.0], [0.05303227890592139]]]], dtype=np.float64)], [np.array([[[[0.125], [0.125], [0.125], [0.125], [0.125], [0.125], [0.125], [0.125]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.125], [0.125], [0.125], [0.125], [0.125], [0.125], [0.125], [0.125]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.125], [0.125], [0.125], [0.125], [0.125], [0.125], [0.125], [0.125]]]], dtype=np.float64)]],
    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    3,
    basix.PolysetType.standard, dtype=np.float64
)

Symfem

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

Examples

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

References

[1] Arnold, Douglas N. and Awanou, Gerard. Finite element differential forms on cubical meshes, Mathematics of computation 83, 1551–5170, 2014. [BibTeX]
[2] 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]
[3] 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]
[4] 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	17 November 2025