Huang–Zhang

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Alternative names	\(Q_{k+1,k}\times Q_{k,k+1}\)
Abbreviated names	HZ
Degrees	\(1\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(2k+1\)
Lagrange subdegree	\(k\)
Lagrange superdegree	\(k+1\)
Reference cells	quadrilateral
Finite dimensional space	\(\mathcal{Z}^{(17)}_{k+1} \oplus \mathcal{Z}^{(18)}_{k+1}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{Z}^{(17)}_k=\operatorname{span}\left\{\left(\begin{array}{c}x^iy^j\\0\end{array}\right)\middle\|i\in\{0,1,...,k+1\}, j\in\{0,1,...,k\}\right\}\) \(\mathcal{Z}^{(18)}_k=\operatorname{span}\left\{\left(\begin{array}{c}0\\x^iy^j\end{array}\right)\middle\|i\in\{0,1,...,k\}, j\in\{0,1,...,k+1\}\right\}\)
DOFs	On each facet: normal integral moments with a degree \(k\) Lagrange space, and tangent integral moments with a degree \(k-1\) Lagrange space On the interior of the reference cell: integral moments with \(\left\{\left(\begin{array}{c}x^iy^j\\0\end{array}\right)\middle\|i\in\{0,1,...,k\}, j\in\{0,1,...,k-1\}\right\}\cup\left\{\left(\begin{array}{c}x^iy^j\\0\end{array}\right)\middle\|i\in\{0,1,...,k-1\}, j\in\{0,1,...,k\}\right\}\)
Number of DOFs	quadrilateral: \(2(k+1)(k+2)\) (A046092)
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 ↓

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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 Huang-Zhang 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], [0.4999999999999998, 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], [0.49999999999999983, 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], [0.2499999999999999, 0.14433756729740635, 0.0, 0.14433756729740632, 0.08333333333333329, 0.0, 6.938893903907228e-18, -2.7755575615628914e-17, -1.3877787807814457e-17, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.33333333333333315, 0.0, -1.3877787807814457e-17, 0.2886751345948127, 0.0, -2.7755575615628914e-17, 0.07453559924999298, 0.0, -2.0816681711721685e-17, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.16666666666666657, 0.09622504486493756, -1.3877787807814457e-17, 0.14433756729740638, 0.0833333333333333, -1.3877787807814457e-17, 0.03726779962499649, 0.02151657414559674, -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, 0.9999999999999996, 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, 0.4999999999999998, 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, 0.33333333333333315, 0.2886751345948127, 0.07453559924999298, 0.0, 0.0, 0.0, -1.3877787807814457e-17, -4.163336342344337e-17, -1.3877787807814457e-17], [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], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.2499999999999999, 0.14433756729740635, 0.0, 0.14433756729740632, 0.08333333333333329, 0.0, 6.938893903907228e-18, -2.7755575615628914e-17, -1.3877787807814457e-17], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.16666666666666657, 0.14433756729740638, 0.03726779962499649, 0.09622504486493758, 0.0833333333333333, 0.021516574145596754, 0.0, -2.7755575615628914e-17, -6.938893903907228e-18]], 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]]], [[[0.5], [0.5]], [[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]]], [[[0.0], [0.0]], [[0.5], [0.5]]]], dtype=np.float64), np.array([[[[-0.39433756729740643], [-0.10566243270259357]], [[0.0], [0.0]]], [[[-0.10566243270259357], [-0.39433756729740643]], [[0.0], [0.0]]], [[[0.0], [0.0]], [[0.5], [0.5]]]], dtype=np.float64), np.array([[[[0.0], [0.0]], [[0.39433756729740643], [0.10566243270259357]]], [[[0.0], [0.0]], [[0.10566243270259357], [0.39433756729740643]]], [[[0.5], [0.5]], [[0.0], [0.0]]]], 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 Huang-Zhang 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], [0.4999999999999997, 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], [0.33333333333333315, 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], [0.49999999999999967, -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], [0.24999999999999983, 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], [0.16666666666666655, 0.14433756729740632, 0.037267799624996364, -1.6653345369377348e-16, 0.09622504486493753, 0.08333333333333322, 0.021516574145596677, -9.194034422677078e-17, -1.0408340855860843e-16, -6.245004513516506e-17, -2.6020852139652106e-17, -1.8214596497756474e-17, -1.249000902703301e-16, -6.938893903907228e-17, -4.163336342344337e-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], [0.33333333333333315, -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], [0.16666666666666657, 0.09622504486493755, -1.0061396160665481e-16, -1.1796119636642288e-16, 0.14433756729740632, 0.08333333333333322, -7.892991815694472e-17, -9.367506770274758e-17, 0.03726779962499635, 0.02151657414559669, -2.7755575615628914e-17, -4.163336342344337e-17, -1.5265566588595902e-16, -6.938893903907228e-17, -3.642919299551295e-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], [0.11111111111111105, 0.09622504486493755, 0.02484519974999757, -1.1362438767648086e-16, 0.09622504486493755, 0.08333333333333323, 0.02151657414559667, -8.890457814381136e-17, 0.024845199749997576, 0.021516574145596677, 0.00555555555555552, -2.688821387764051e-17, -1.1796119636642288e-16, -7.632783294297951e-17, -2.949029909160572e-17, -1.3010426069826053e-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], [0.24999999999999978, -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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.1249999999999999, 0.07216878364870313, -7.37257477290143e-17, -8.673617379884035e-17, 0.12990381056766565, 0.0749999999999999, -7.719519468096792e-17, -8.066464163292153e-17, 0.05590169943749458, 0.032274861218395054, -3.8163916471489756e-17, -4.0766001685454967e-17, 0.009449111825230519, 0.005455447255899734, -2.0816681711721685e-17, -2.0816681711721685e-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], [0.08333333333333326, 0.07216878364870315, 0.018633899812498175, -9.020562075079397e-17, 0.08660254037844378, 0.07499999999999989, 0.019364916731036998, -9.020562075079397e-17, 0.037267799624996406, 0.032274861218395054, 0.008333333333333281, -4.640385298237959e-17, 0.006299407883487006, 0.005455447255899731, 0.0014085904245474887, -1.734723475976807e-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, 0.9999999999999996, -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, 0.4999999999999997, 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, 0.33333333333333315, 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, 0.24999999999999983, 0.2598076211353313, 0.11180339887498922, 0.018898223650461014, -2.0816681711721685e-17, -2.7755575615628914e-17, -1.1275702593849246e-17, 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    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    3,
    basix.PolysetType.standard, dtype=np.float64
)

Symfem

"HZ"
↓ 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)

References

[1] Zhang, Shangyou. A family of \(Q_{k+1,k}\times Q+{k,k+1}\) divergence-free finite elements on rectangular grids, SIAM journal on numerical analysis 47(3), 2090–2107, 2009. [DOI: 10.1137/080728949] [BibTeX]
[2] Huang, Yunqing and Zhang, Shangyou. A lowest order divergence-free finite element on rectangular grids, Frontiers of mathematics in China 6, 253–270, 2011. [DOI: 10.1007/s11464-011-0094-0] [BibTeX]

DefElement stats

Element added	09 December 2022
Element last updated	04 June 2025