Tiniest tensor H(div)

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Alternative names	TNT H(div)
De Rham complex families	\(\left[S_{3,k}^\square\right]_{d-1}\)
Degrees	\(1\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(dk+1\)
Lagrange subdegree	\(k\)
Lagrange superdegree	\(k+1\)
Reference cells	quadrilateral, hexahedron
Finite dimensional space	\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(38)}_{k}\) (quadrilateral) \(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(39)}_{k}\) (hexahedron) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)\leqslant k\right\}\) \(\mathcal{Z}^{(38)}_k=\left\{\left(\begin{array}{c}\tilde{B}_{k+1}(x)\tilde{P}_{k}(y)\\\tilde{P}_k(x)\tilde{B}_{k+1}(y)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(0)\tilde{P}_{k}(y)\\\tilde{P}_k(0)\tilde{B}_{k+1}(y)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\\\tilde{P}_k(x)\tilde{B}_{k+1}(0)\end{array}\right)\right\} \) where \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\) \(\tilde{B}_k\) is \((\tilde{P}_k-\tilde{P}_{k-2})/(4k-2)\) \(\mathcal{Z}^{(39)}_k=\left\{\left(\begin{array}{c}\tilde{B}_{k+1}(x)\tilde{P}_{k}(y)\tilde{P}_{k}(z)\\\tilde{P}_k(x)\tilde{B}_{k+1}(y)\tilde{P}_{k}(z)\\\tilde{P}_{k}(x)\tilde{P}_{k}(y)\tilde{B}_{k+1}(z)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(x)\tilde{P}_{k}(y)\tilde{P}_{k}(0)\\\tilde{P}_k(x)\tilde{B}_{k+1}(y)\tilde{P}_{k}(0)\\\tilde{P}_{k}(x)\tilde{P}_{k}(y)\tilde{B}_{k+1}(0)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\tilde{P}_{k}(z)\\\tilde{P}_k(x)\tilde{B}_{k+1}(0)\tilde{P}_{k}(z)\\\tilde{P}_{k}(x)\tilde{P}_{k}(0)\tilde{B}_{k+1}(z)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\tilde{P}_{k}(0)\\\tilde{P}_k(x)\tilde{B}_{k+1}(0)\tilde{P}_{k}(0)\\\tilde{P}_{k}(x)\tilde{P}_{k}(0)\tilde{B}_{k+1}(0)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(0)\tilde{P}_{k}(y)\tilde{P}_{k}(z)\\\tilde{P}_k(0)\tilde{B}_{k+1}(y)\tilde{P}_{k}(z)\\\tilde{P}_{k}(0)\tilde{P}_{k}(y)\tilde{B}_{k+1}(z)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(0)\tilde{P}_{k}(y)\tilde{P}_{k}(0)\\\tilde{P}_k(0)\tilde{B}_{k+1}(y)\tilde{P}_{k}(0)\\\tilde{P}_{k}(0)\tilde{P}_{k}(y)\tilde{B}_{k+1}(0)\end{array}\right),\left(\begin{array}{c}\tilde{B}_{k+1}(0)\tilde{P}_{k}(0)\tilde{P}_{k}(z)\\\tilde{P}_k(0)\tilde{B}_{k+1}(0)\tilde{P}_{k}(z)\\\tilde{P}_{k}(0)\tilde{P}_{k}(0)\tilde{B}_{k+1}(z)\end{array}\right)\right\} \) where \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\) \(\tilde{B}_k\) is \((\tilde{P}_k-\tilde{P}_{k-2})/(4k-2)\)
DOFs	On each facet: normal integral moments with a degree \(k\) Lagrange space On the interior of the reference cell: integral moments with \(\nabla f\) for each \(f\) in a degree \(k\) Lagrange space, and integral moments with \(\nabla\times\boldsymbol{f}\) for each \(\boldsymbol{f}\) in a degree \(k\) vector Lagrange space such that the tangential trace of \(\boldsymbol{f}\) on the facets of the cell is 0
Number of DOFs	quadrilateral: \(2(k+1)^2 + 3\) hexahedron: \(3(k+1)^3 + 7\)
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 Tiniest tensor 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], [0.4999999999999998, 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], [0.49999999999999983, 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], [0.2499999999999999, 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], [0.0, 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, 0.2499999999999999, -2.949029909160572e-17, -0.11180339887498948, -5.204170427930421e-18, 0.0, 1.734723475976807e-18, 1.734723475976807e-18, -3.469446951953614e-18, -1.3877787807814457e-17], [0.2499999999999999, 8.673617379884035e-18, 1.734723475976807e-18, -3.2959746043559335e-17, 0.0, 3.469446951953614e-18, -0.11180339887498948, 3.469446951953614e-18, -1.3877787807814457e-17, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, -0.14433756729740638, 0.0, 3.469446951953614e-18, 2.0816681711721685e-17, 1.734723475976807e-18, -3.469446951953614e-18, 0.06454972243679029, 8.673617379884035e-19, 0.0, 0.0, 0.0, -0.14433756729740638, 1.3877787807814457e-17, 0.06454972243679027, 3.469446951953614e-18, 0.0, 8.673617379884035e-19]], 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.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]], [[0.39433756729740643], [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.array([[[[0.0], [0.0], [0.0], [0.0]], [[0.25], [0.25], [0.25], [0.25]]], [[[0.25], [0.25], [0.25], [0.25]], [[0.0], [0.0], [0.0], [0.0]]], [[[0.052831216351296784], [0.19716878364870322], [0.052831216351296784], [0.19716878364870322]], [[0.052831216351296784], [0.052831216351296784], [0.19716878364870322], [0.19716878364870322]]]], dtype=np.float64)], []],
    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    2,
    basix.PolysetType.standard, dtype=np.float64
)

# Create Tiniest tensor 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], [0.4999999999999997, 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], [0.33333333333333315, 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], [0.49999999999999967, -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], [0.24999999999999983, 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], [0.16666666666666655, 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], [0.0, 0.0, 0.0, 0.0, 0.0, 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], [0.33333333333333315, -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], [0.16666666666666657, 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], [0.0, 0.0, 0.0, 0.0, 0.0, 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], [0.11111111111111105, 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], [0.0, 0.0, 0.0, 0.0, 0.0, 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, -9.71445146547012e-17, -0.11547005383792525, -8.153200337090993e-17, 0.07559289460184551, 0.0, 1.3010426069826053e-17, 0.0, -5.204170427930421e-18, -2.6020852139652106e-18, 3.903127820947816e-17, -8.673617379884035e-19, -2.7755575615628914e-17, 0.0, 1.3877787807814457e-17, 6.938893903907228e-18, 1.3877787807814457e-17], [-9.020562075079397e-17, 1.734723475976807e-18, -2.6020852139652106e-18, 2.6020852139652106e-18, -0.11547005383792525, 1.3877787807814457e-17, 4.683753385137379e-17, 1.9081958235744878e-17, -8.326672684688674e-17, -3.469446951953614e-18, 3.469446951953614e-18, 0.0, 0.07559289460184551, 0.0, -3.469446951953614e-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], [8.673617379884035e-17, 2.211772431870429e-17, -5.0306980803327406e-17, 6.938893903907228e-18, -1.9081958235744878e-17, -1.5612511283791264e-17, -0.051639777949432246, 2.949029909160572e-17, -7.979727989493313e-17, -2.6020852139652106e-17, -2.7755575615628914e-17, 3.469446951953614e-18, 6.418476861114186e-17, 3.382710778154774e-17, 0.033806170189140665, -1.9081958235744878e-17, -1.2706849461530112e-16, -1.1145598333150986e-16, -6.158268339717665e-17, -3.903127820947816e-17, -1.8735013540549517e-16, -1.6653345369377348e-16, -4.85722573273506e-17, 3.469446951953614e-18, -3.885780586188048e-16, -0.051639777949432523, -9.367506770274758e-17, 0.03380617018914067, -4.974319567363494e-16, -3.916138247017642e-16, -1.1622647289044608e-16, -1.6479873021779667e-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], [0.1127016653792583, 0.1127016653792583], [0.1127016653792583, 0.5], [0.1127016653792583, 0.8872983346207417], [0.5, 0.1127016653792583], [0.5, 0.5], [0.5, 0.8872983346207417], [0.8872983346207417, 0.1127016653792583], [0.8872983346207417, 0.5], [0.8872983346207417, 0.8872983346207417]], 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([[[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.25], [0.25], [0.25], [0.25], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.017392232311613923], [0.12345679012345674], [0.13692875534270696], [0.027827571698582285], [0.19753086419753085], [0.21908600854833118], [0.017392232311613923], [0.12345679012345674], [0.13692875534270696]]], [[[0.25], [0.25], [0.25], [0.25], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.008696116155806961], [0.06172839506172837], [0.06846437767135348], [0.013913785849291142], [0.09876543209876543], [0.10954300427416559], [0.008696116155806961], [0.06172839506172837], [0.06846437767135348]], [[0.0], [0.0], [0.0], [0.0], [0.008696116155806961], [0.013913785849291142], [0.008696116155806961], [0.06172839506172837], [0.09876543209876543], [0.06172839506172837], [0.06846437767135348], [0.10954300427416559], [0.06846437767135348]]], [[[0.0], [0.0], [0.0], [0.0], [0.000980066773090918], [0.030864197530864185], [0.06074832828863743], [0.0015681068369454694], [0.04938271604938271], [0.09719732526181993], [0.000980066773090918], [0.030864197530864185], [0.06074832828863743]], [[0.0], [0.0], [0.0], [0.0], [0.001960133546181836], [0.013913785849291142], [0.015432098765432086], [0.013913785849291142], [0.09876543209876543], [0.10954300427416559], [0.015432098765432086], [0.10954300427416559], [0.12149665657727486]]], [[[0.0], [0.0], [0.0], [0.0], [0.017392232311613923], [0.027827571698582285], [0.017392232311613923], [0.12345679012345674], [0.19753086419753085], [0.12345679012345674], [0.13692875534270696], [0.21908600854833118], [0.13692875534270696]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.001960133546181836], [0.013913785849291142], [0.015432098765432086], [0.013913785849291142], [0.09876543209876543], [0.10954300427416559], [0.015432098765432086], [0.10954300427416559], [0.12149665657727486]], [[0.0], [0.0], [0.0], [0.0], [0.000980066773090918], [0.0015681068369454694], [0.000980066773090918], [0.030864197530864185], [0.04938271604938271], [0.030864197530864185], [0.06074832828863743], [0.09719732526181993], [0.06074832828863743]]], [[[0.0], [0.0], [0.0], [0.0], [0.00022091031502044424], [0.006956892924645571], [0.013692875534270694], [0.0015681068369454694], [0.04938271604938271], [0.09719732526181993], [0.001739223231161392], [0.054771502137082796], [0.10780378104300417]], [[0.0], [0.0], [0.0], [0.0], [0.00022091031502044424], [0.0015681068369454694], [0.001739223231161392], [0.006956892924645571], [0.04938271604938271], [0.054771502137082796], [0.013692875534270694], [0.09719732526181993], [0.10780378104300417]]], [[[0.0], [0.0], [0.0], [0.0], [0.005976826151554651], [0.0], [-0.005976826151554651], [0.023907304606218614], [0.0], [-0.023907304606218614], [0.005976826151554651], [0.0], [-0.005976826151554651]], [[0.0], [0.0], [0.0], [0.0], [-0.005976826151554651], [-0.023907304606218614], [-0.005976826151554651], [0.0], [0.0], [0.0], [0.005976826151554651], [0.023907304606218614], [0.005976826151554651]]]], dtype=np.float64)], []],
    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    3,
    basix.PolysetType.standard, dtype=np.float64
)

# Create Tiniest tensor H(div) degree 3 on a quadrilateral
e = basix.ufl.custom_element(
    basix.CellType.quadrilateral,
    (2, ),
    np.array([[1.0000000000000002, -8.673617379884035e-17, -1.3530843112619095e-16, -1.734723475976807e-16, -6.696032617270475e-16, -8.673617379884035e-17, 0.0, 6.938893903907228e-18, 1.734723475976807e-17, -1.734723475976807e-17, -1.3183898417423734e-16, -2.7755575615628914e-17, 6.938893903907228e-17, 2.0816681711721685e-17, -4.163336342344337e-17, -1.457167719820518e-16, 2.7755575615628914e-17, 1.3877787807814457e-17, 0.0, -3.8163916471489756e-17, -6.83481049534862e-16, 1.3877787807814457e-17, -6.245004513516506e-17, -2.2551405187698492e-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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0000000000000002, -8.673617379884035e-17, -1.3530843112619095e-16, -1.734723475976807e-16, -6.696032617270475e-16, -8.673617379884035e-17, 0.0, 6.938893903907228e-18, 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[0.0051459595190854225], [0.043061891446125065], [0.061543964721302025], [2.6304488021623317e-05], [0.005295161448637261], [0.04431042775312181], [0.06332837018633383]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [1.464370196598415e-07], [6.2020108504632095e-06], [2.556330564092189e-05], [2.630448802162332e-05], [2.947815066838457e-05], [0.0012484808194102267], [0.005145959519085423], [0.005295161448637261], [0.0002466760415441627], [0.010447409335152993], [0.043061891446125065], [0.04431042775312181], [0.0003525488799621802], [0.014931415457123759], [0.061543964721302025], [0.06332837018633383]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0016831193488865217], [0.0012457873822048421], [-0.0012457873822048413], [-0.0016831193488865217], [0.010798148914104612], [0.007992420547752498], [-0.007992420547752494], [-0.01079814891410461], [0.01079814891410461], [0.007992420547752494], [-0.007992420547752493], [-0.010798148914104608], [0.0016831193488865233], [0.0012457873822048428], [-0.001245787382204843], [-0.001683119348886523]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [-0.001683119348886522], [-0.010798148914104612], [-0.01079814891410461], [-0.0016831193488865233], [-0.0012457873822048421], [-0.007992420547752498], [-0.007992420547752494], [-0.001245787382204843], [0.0012457873822048417], [0.007992420547752494], [0.007992420547752493], [0.0012457873822048428], [0.0016831193488865217], [0.01079814891410461], [0.010798148914104608], [0.001683119348886523]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.00024314653640150074], [0.0012213068291441814], [-2.4480553060660155e-05], [-0.0014399728124850212], [0.0015599205782698062], [0.007835364152658751], [-0.00015705639509374465], [-0.009238228335834806], [0.0015599205782698058], [0.007835364152658748], [-0.00015705639509374457], [-0.009238228335834804], [0.00024314653640150095], [0.0012213068291441825], [-2.448055306066018e-05], [-0.0014399728124850223]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [-0.00011686208040689954], [-0.0035634914887513213], [-0.007234657425353288], [-0.0015662572684796235], [-8.649731543127707e-05], [-0.0026375745345792773], [-0.005354846013173219], [-0.001159290066773566], [8.649731543127704e-05], [0.0026375745345792764], [0.005354846013173217], [0.0011592900667735657], [0.00011686208040689952], [0.0035634914887513205], [0.007234657425353288], [0.0015662572684796232]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.00011686208040689956], [8.649731543127707e-05], [-8.649731543127704e-05], [-0.00011686208040689956], [0.003563491488751322], [0.0026375745345792773], [-0.002637574534579277], [-0.003563491488751322], [0.007234657425353288], [0.0053548460131732185], [-0.005354846013173218], [-0.007234657425353288], [0.0015662572684796237], [0.001159290066773566], [-0.0011592900667735662], [-0.0015662572684796232]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [-0.00024314653640150074], [-0.001559920578269806], [-0.0015599205782698055], [-0.00024314653640150095], [-0.0012213068291441816], [-0.007835364152658751], [-0.007835364152658748], [-0.0012213068291441825], [2.4480553060660158e-05], [0.00015705639509374462], [0.00015705639509374457], [2.448055306066018e-05], [0.0014399728124850214], [0.009238228335834806], [0.009238228335834804], [0.0014399728124850225]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [1.6882112433921675e-05], [8.479758548516665e-05], [-1.6997299461103875e-06], [-9.997996797297788e-05], [0.0005147885760800725], [0.0025857444355852277], [-5.183009899404892e-05], [-0.0030487029126712497], [0.0010451320021897335], [0.005249619717073521], [-0.00010522629609969568], [-0.006189525423163556], [0.00022626442396757928], [0.0011365092436590157], [-2.278082311454979e-05], [-0.0013399928445120442]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [0.0], [-1.6882112433921675e-05], [-0.0005147885760800725], [-0.0010451320021897333], [-0.00022626442396757922], [-8.479758548516665e-05], [-0.0025857444355852272], [-0.005249619717073522], [-0.0011365092436590157], [1.6997299461103875e-06], [5.183009899404891e-05], [0.00010522629609969568], [2.278082311454979e-05], [9.997996797297787e-05], [0.0030487029126712497], [0.006189525423163555], [0.0013399928445120444]]]], dtype=np.float64)], []],
    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    4,
    basix.PolysetType.standard, dtype=np.float64
)

# Create Tiniest tensor H(div) 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, 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, 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], [0.49999999999999967, 0.28867513459481264, -6.938893903907228e-18, 1.3877787807814457e-17, 6.938893903907228e-18, 6.938893903907228e-18, -4.163336342344337e-17, 1.0408340855860843e-17, -4.5102810375396984e-17, 0.0, 6.938893903907228e-18, 0.0, -6.938893903907228e-18, 0.0, 0.0, -3.469446951953614e-18, -1.3877787807814457e-17, 0.0, -1.3877787807814457e-17, 1.0408340855860843e-17, -5.204170427930421e-17, 6.938893903907228e-18, 6.938893903907228e-18, -3.469446951953614e-18, -4.5102810375396984e-17, 2.42861286636753e-17, -3.642919299551295e-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, 0.49999999999999967, 0.28867513459481264, -6.938893903907228e-18, 1.3877787807814457e-17, 6.938893903907228e-18, 6.938893903907228e-18, -4.163336342344337e-17, 1.0408340855860843e-17, -4.5102810375396984e-17, 0.0, 6.938893903907228e-18, 0.0, -6.938893903907228e-18, 0.0, 0.0, -3.469446951953614e-18, -1.3877787807814457e-17, 0.0, -1.3877787807814457e-17, 1.0408340855860843e-17, -5.204170427930421e-17, 6.938893903907228e-18, 6.938893903907228e-18, -3.469446951953614e-18, -4.5102810375396984e-17, 2.42861286636753e-17, -3.642919299551295e-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, 0.49999999999999967, 0.28867513459481264, -6.938893903907228e-18, 1.3877787807814457e-17, 6.938893903907228e-18, 6.938893903907228e-18, -4.163336342344337e-17, 1.0408340855860843e-17, -4.5102810375396984e-17, 0.0, 6.938893903907228e-18, 0.0, -6.938893903907228e-18, 0.0, 0.0, -3.469446951953614e-18, -1.3877787807814457e-17, 0.0, -1.3877787807814457e-17, 1.0408340855860843e-17, -5.204170427930421e-17, 6.938893903907228e-18, 6.938893903907228e-18, -3.469446951953614e-18, -4.5102810375396984e-17, 2.42861286636753e-17, -3.642919299551295e-17], [0.49999999999999967, 0.0, -2.0816681711721685e-17, 0.28867513459481264, 0.0, -3.469446951953614e-18, -6.938893903907228e-18, 0.0, -3.469446951953614e-17, 1.3877787807814457e-17, -6.938893903907228e-18, 0.0, 0.0, 0.0, -3.469446951953614e-18, 1.0408340855860843e-17, -1.3877787807814457e-17, 0.0, -6.938893903907228e-18, 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    0,
    basix.MapType.contravariantPiola,
    basix.SobolevSpace.HDiv,
    False,
    -1,
    2,
    basix.PolysetType.standard, dtype=np.float64
)

Symfem

"TNTdiv"
↓ 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)
quadrilateral degree 3	(click to view basis functions)
hexahedron degree 1	(click to view basis functions)

References

[1] Cockburn, Bernardo and Qiu, Weifeng. Commuting diagrams for the TNT elements on cubes, Mathematics of Computation 83, 603–633, 2014. [DOI: 10.1090/S0025-5718-2013-02729-9] [BibTeX]
[2] 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	24 October 2021
Element last updated	04 June 2025