Conforming Crouzeix–Raviart

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Abbreviated names	conforming CR
Degrees	\(1\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Reference cells	triangle
Finite dimensional space	\(\mathcal{P}_{k} \oplus \mathcal{Z}^{(12)}_{k}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(12)}_k=\left\{x^iy^{k-i}(x+y)\middle\|i=1,...,k-1\right\}\)
DOFs	On each vertex: points evaluation On each edge: point evaluations On each face: point evaluations
Number of DOFs	triangle: \(k(k+5)/2\)
Mapping	identity
continuity	Discontinuous.
Categories	Scalar-valued 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 conforming Crouzeix-Raviart degree 1 on a triangle
e = basix.ufl.custom_element(
    basix.CellType.triangle,
    (),
    np.array([[0.7071067811865475, -5.551115123125783e-17, -1.6653345369377348e-16], [0.23570226039551584, -0.08333333333333334, 0.14433756729740632], [0.2357022603955158, 0.16666666666666657, -6.245004513516506e-17]], dtype=np.float64),
    [[np.array([[0.0, 0.0]], dtype=np.float64), np.array([[1.0, 0.0]], dtype=np.float64), np.array([[0.0, 1.0]], 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([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64)], [np.empty((0, 1, 0, 1), dtype=np.float64), np.empty((0, 1, 0, 1), dtype=np.float64), np.empty((0, 1, 0, 1), dtype=np.float64)], [np.empty((0, 1, 0, 1), dtype=np.float64)], []],
    0,
    basix.MapType.identity,
    basix.SobolevSpace.L2,
    False,
    -1,
    1,
    basix.PolysetType.standard, dtype=np.float64
)

# Create conforming Crouzeix-Raviart degree 2 on a triangle
e = basix.ufl.custom_element(
    basix.CellType.triangle,
    (),
    np.array([[0.7071067811865489, -3.608224830031759e-16, -5.967448757360216e-16, 3.434752482434078e-15, -4.926614671774132e-16, 2.67841304690819e-15, -3.2335245592207684e-15, 3.191891195797325e-16, 7.896461262646426e-15, 4.649058915617843e-16], [0.23570226039551628, -0.08333333333333401, 0.14433756729740724, 1.40989650510015e-15, 3.0357660829594124e-16, 6.557254739192331e-16, -1.7403613272737317e-15, 7.485331798839923e-16, 2.4060614611798314e-15, 1.0174153186603974e-15], [0.11785113019775828, -0.06666666666666712, 0.11547005383792579, 0.01360827634879615, -0.023570226039551643, 0.030429030972509735, -9.993633524885137e-16, 5.049671618351237e-16, 1.0830908652587068e-15, 6.541533807691291e-16], [0.23570226039551628, 0.1666666666666677, -2.0469737016526324e-16, 7.667477763817487e-16, -2.255140518769849e-16, 1.349614864309956e-15, 1.4051260155412137e-16, 5.898059818321144e-17, 3.226585665316861e-15, 1.6306400674181987e-16], [0.05892556509887893, 0.016666666666666784, 0.028867513459481495, -0.020412414523193062, 0.02357022603955198, 3.8510861166685117e-16, -1.0592655225183378e-16, 2.205267218835516e-16, 8.020927672047762e-16, 3.2482697087665713e-16], [0.1178511301977583, 0.1333333333333341, -7.632783294297951e-17, 0.04082482904638698, -1.0495077029659683e-16, 5.872038966181492e-16, 3.616898447411643e-16, 5.204170427930421e-18, 1.6705387073656652e-15, 5.941427905220564e-17], [0.04714045207910321, 0.016666666666666795, 0.028867513459481513, -0.015552315827194733, 0.02357022603955199, 0.004347004424644534, -0.0016835875742537718, 2.1217836515541322e-16, 0.003764616262105953, 3.1417468453198705e-16]], dtype=np.float64),
    [[np.array([[0.0, 0.0]], dtype=np.float64), np.array([[1.0, 0.0]], dtype=np.float64), np.array([[0.0, 1.0]], dtype=np.float64)], [np.array([[0.5, 0.5]], dtype=np.float64), np.array([[0.0, 0.5]], dtype=np.float64), np.array([[0.5, 0.0]], dtype=np.float64)], [np.array([[0.3333333333333333, 0.3333333333333333]], dtype=np.float64)], []],
    [[np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64)], [np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64)], [np.array([[[[1.0]]]], dtype=np.float64)], []],
    0,
    basix.MapType.identity,
    basix.SobolevSpace.L2,
    False,
    -1,
    3,
    basix.PolysetType.standard, dtype=np.float64
)

# Create conforming Crouzeix-Raviart degree 3 on a triangle
e = basix.ufl.custom_element(
    basix.CellType.triangle,
    (),
    np.array([[0.7071067811865476, 0.0, 2.7755575615628914e-17, -3.885780586188048e-16, 3.469446951953614e-17, -3.0531133177191805e-16, 2.636779683484747e-16, 4.163336342344337e-17, -3.0531133177191805e-16, 9.020562075079397e-17, -4.440892098500626e-16, 1.3877787807814457e-17, -3.191891195797325e-16, 1.249000902703301e-16, -4.0245584642661925e-16], [0.23570226039551584, -0.08333333333333329, 0.14433756729740638, -1.3183898417423734e-16, 3.469446951953614e-17, -7.28583859910259e-17, 1.5265566588595902e-16, -7.632783294297951e-17, -4.163336342344337e-17, -9.71445146547012e-17, -1.249000902703301e-16, 9.71445146547012e-17, -7.632783294297951e-17, 4.683753385137379e-17, -1.249000902703301e-16], [0.11785113019775792, -0.06666666666666662, 0.1154700538379251, 0.01360827634879535, -0.023570226039551542, 0.030429030972509173, 1.0408340855860843e-16, -7.632783294297951e-17, 4.163336342344337e-17, -1.0408340855860843e-16, -9.71445146547012e-17, 9.020562075079397e-17, -7.979727989493313e-17, 7.025630077706069e-17, -1.0755285551056204e-16], [0.07071067811865474, -0.04999999999999997, 0.08660254037844381, 0.017496355305594073, -0.03030457633656627, 0.039123039821797524, -0.00252538136138045, 0.004374088826398458, -0.005646924393157764, 0.006681531047810515, -8.326672684688674e-17, 7.632783294297951e-17, -7.806255641895632e-17, 7.979727989493313e-17, -1.0408340855860843e-16], [0.2357022603955159, 0.16666666666666657, 1.0408340855860843e-17, -9.71445146547012e-17, 3.122502256758253e-17, -8.326672684688674e-17, -7.632783294297951e-17, 3.469446951953614e-17, -1.734723475976807e-16, 2.5153490401663703e-17, -1.8041124150158794e-16, 1.734723475976807e-17, -1.457167719820518e-16, 5.984795992119984e-17, -1.5178830414797062e-17], [0.058925565098878974, 0.016666666666666663, 0.028867513459481277, -0.020412414523193156, 0.02357022603955157, -1.6479873021779667e-17, 1.0408340855860843e-17, -7.806255641895632e-18, -3.209238430557093e-17, -3.0357660829594124e-18, -3.469446951953614e-18, 1.734723475976807e-17, -2.2551405187698492e-17, -1.6479873021779667e-17, 8.239936510889834e-18], [0.02357022603955159, -3.903127820947816e-18, 0.01924500897298752, -0.00972019739199674, 0.0101015254455221, 0.00434700442464417, 0.003367175148507377, -0.004860098695998375, 0.003764616262105204, -4.336808689942018e-19, -1.734723475976807e-18, 8.673617379884035e-18, -1.3877787807814457e-17, -5.204170427930421e-18, 6.5052130349130266e-18], [0.1178511301977579, 0.13333333333333328, 5.204170427930421e-18, 0.040824829046386214, 1.734723475976807e-17, -3.382710778154774e-17, -1.0755285551056204e-16, 1.214306433183765e-17, -8.326672684688674e-17, 9.64939933512099e-18, -1.8041124150158794e-16, 6.938893903907228e-18, -8.847089727481716e-17, 3.426078865054194e-17, -4.9873299934333204e-18], [0.02357022603955159, 0.01666666666666667, 0.009622504486493759, -0.005832118435198043, 0.013468700594029468, -6.5052130349130266e-18, -0.005050762722761049, 0.0048600986959983685, -1.5395670849294163e-17, -2.0599841277224584e-18, -3.469446951953614e-18, 8.673617379884035e-18, -5.204170427930421e-18, -9.107298248878237e-18, 2.3852447794681098e-18], [0.07071067811865474, 0.09999999999999992, 3.903127820947816e-18, 0.05248906591678231, 1.214306433183765e-17, -1.452830911130576e-17, 0.010101525445521991, 8.998878031629687e-18, -4.5102810375396984e-17, 4.662069341687669e-18, -1.8388068845354155e-16, 5.204170427930421e-18, -5.377642775528102e-17, 2.2551405187698492e-17, -4.119968255444917e-18], [0.01964185503295966, 0.015079365079365078, 0.009622504486493759, -0.0038880789567986938, 0.013468700594029468, 0.0010867511061610387, -0.004676632150704676, 0.004860098695998369, 0.0016731627831578597, -8.673617379884035e-19, -0.00041829069578947463, 8.673617379884035e-18, 0.0006547285010986498, -9.107298248878237e-18, 2.6020852139652106e-18], [0.019641855032959656, 0.00079365079365079, 0.017870365474916987, -0.008262167783197228, 0.009680628551958679, 0.005977131083885739, 0.002338316075352348, -0.004374088826398537, 0.004810343001578874, 0.0007423923386456238, 0.0002091453478947321, -0.0001086751106160952, -0.0003273642505493427, 0.0006640158940746585, 7.589415207398531e-18]], dtype=np.float64),
    [[np.array([[0.0, 0.0]], dtype=np.float64), np.array([[1.0, 0.0]], dtype=np.float64), np.array([[0.0, 1.0]], dtype=np.float64)], [np.array([[0.6666666666666666, 0.3333333333333333], [0.3333333333333333, 0.6666666666666666]], dtype=np.float64), np.array([[0.0, 0.3333333333333333], [0.0, 0.6666666666666666]], dtype=np.float64), np.array([[0.3333333333333333, 0.0], [0.6666666666666666, 0.0]], dtype=np.float64)], [np.array([[0.2222222222222222, 0.2222222222222222], [0.2222222222222222, 0.5555555555555556], [0.5555555555555556, 0.2222222222222222]], dtype=np.float64)], []],
    [[np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64)], [np.array([[[[1.0], [0.0]]], [[[0.0], [1.0]]]], dtype=np.float64), np.array([[[[1.0], [0.0]]], [[[0.0], [1.0]]]], dtype=np.float64), np.array([[[[1.0], [0.0]]], [[[0.0], [1.0]]]], dtype=np.float64)], [np.array([[[[1.0], [0.0], [0.0]]], [[[0.0], [1.0], [0.0]]], [[[0.0], [0.0], [1.0]]]], dtype=np.float64)], []],
    0,
    basix.MapType.identity,
    basix.SobolevSpace.L2,
    False,
    -1,
    4,
    basix.PolysetType.standard, dtype=np.float64
)

# Create conforming Crouzeix-Raviart degree 4 on a triangle
e = basix.ufl.custom_element(
    basix.CellType.triangle,
    (),
    np.array([[0.7071067811865477, -2.7755575615628914e-17, -1.0408340855860843e-17, -1.1709383462843448e-16, -2.7755575615628914e-17, -9.71445146547012e-17, 1.1796119636642288e-16, 0.0, -2.220446049250313e-16, 6.250425524378933e-17, 1.0061396160665481e-16, 6.613633252161577e-17, -6.938893903907228e-17, -1.0451708942760263e-16, 3.913969842672671e-17, -6.591949208711867e-17, 6.245004513516506e-17, 1.3877787807814457e-16, -1.2663481374630692e-16, 9.107298248878237e-17, 1.3097162243624894e-16], [0.23570226039551587, -0.08333333333333333, 0.1443375672974064, -3.812325889002155e-17, -1.452830911130576e-17, -1.7997756063259374e-17, 3.0791341698588326e-17, -2.862293735361732e-17, -8.370040771588094e-17, 2.9449641510137514e-17, 1.2576745200831851e-17, 1.0869126779167182e-17, -4.5319650809894085e-17, -7.579928438389283e-17, 4.7196675821009615e-17, 8.890457814381136e-18, -4.336808689942018e-19, 1.1796119636642288e-16, -1.0383946306979919e-16, 5.2231439659489176e-17, -3.411848711540322e-17], [0.11785113019775793, -0.06666666666666665, 0.11547005383792512, 0.013608276348795413, -0.02357022603955158, 0.030429030972509246, 2.4482640307438297e-17, -1.5341460740669888e-17, -3.6625704639275947e-17, 3.635211299731281e-17, -5.905513708256982e-18, 9.727855761860419e-18, -2.6007299612496038e-17, -3.6955623972231497e-17, 2.8150081210812604e-17, 1.2685165418080402e-17, -1.555830117516699e-17, 8.30431101488116e-17, -6.772960149540111e-17, 5.337662820417699e-17, -4.166862116987283e-17], [0.07071067811865474, -0.049999999999999996, 0.08660254037844385, 0.01749635530559412, -0.030304576336566323, 0.039123039821797594, -0.0025253813613805164, 0.0043740888263985225, -0.005646924393157833, 0.006681531047810633, -1.2439525863376655e-17, 1.0845105454178642e-17, -1.974306745107686e-17, -9.245060216815269e-18, 6.427504379447345e-18, 1.809876474221945e-17, -1.7526382228112355e-17, 5.5508610132416064e-17, -6.552567470620462e-17, 6.548184075118421e-17, -4.657280958557722e-17], [0.04714045207910317, -0.03809523809523809, 0.06598288790738581, 0.01749635530559412, -0.03030457633656632, 0.03912303982179759, -0.004489566864676482, 0.007776157913597381, -0.01003897669894724, 0.011878277418329988, 0.0005019488349473457, -0.00086940088492882, 0.0011223917161691, -0.0013280317881493196, 0.0015058465048420787, 2.0597849426309557e-17, -2.402468665054934e-17, 5.242930126201186e-17, -6.754861272301328e-17, 6.945688530894861e-17, -6.042316105324421e-17], [0.23570226039551587, 0.16666666666666669, 0.0, -3.664603343001005e-17, -1.3877787807814457e-17, -4.0766001685454967e-17, 5.204170427930421e-17, 1.3877787807814457e-17, -9.367506770274758e-17, -3.74049749507499e-18, 5.204170427930421e-17, 4.195862407518902e-17, -3.122502256758253e-17, -4.0332320816460765e-17, 1.8431436932253575e-18, -8.326672684688674e-17, 6.245004513516506e-17, -1.0408340855860843e-17, -5.160802341031001e-17, -4.336808689942018e-19, 1.2576745200831851e-17], [0.05892556509887897, 0.016666666666666663, 0.028867513459481284, -0.020412414523193156, 0.02357022603955157, -1.4609624274242172e-17, 9.432558900623889e-18, -9.75781955236954e-18, -3.69712940817557e-17, -2.9459805905504566e-18, 1.6534083130403943e-17, -6.120660076859574e-18, -4.336808689942018e-18, -3.341375570328764e-17, 5.692061405548898e-18, 5.854691731421724e-18, 9.974659986866641e-18, 6.396792817664476e-18, -1.8079071226195786e-17, -8.565197162635485e-18, 8.05020113070487e-18], [0.023570226039551584, -1.6601845766184287e-18, 0.01924500897298752, -0.009720197391996739, 0.010101525445522098, 0.00434700442464417, 0.00336717514850737, -0.004860098695998369, 0.003764616262105197, -4.182754572660142e-19, 3.5067164016328034e-18, -1.1064897269101332e-18, -2.066760391300493e-18, -1.8599149455809927e-17, 4.9892358175646426e-18, 2.507217523872729e-19, 4.4858864886587746e-18, 7.508100044462118e-18, -8.881140451961339e-18, -7.601273668660091e-18, 6.481178475034686e-18], [0.011785113019775792, -0.002380952380952382, 0.012371791482634837, -0.004374088826398532, 0.0037880720420707856, 0.004890379977724697, 0.0030865772194650877, -0.00437408882639853, 0.0031371802184210025, 0.0007423923386456252, -0.0006274360436842015, 0.0009780759955449378, -0.0009820927516479819, 0.0006640158940746514, 4.959056960096258e-18, -8.470329472543003e-21, 2.6012381810179563e-18, 5.0849505406043785e-18, -3.880231461592348e-18, -3.3418096747142318e-18, 5.572196317345064e-18], [0.11785113019775792, 0.13333333333333333, 0.0, 0.04082482904638629, 0.0, -1.734723475976807e-17, 2.6020852139652106e-17, 1.734723475976807e-17, -3.642919299551295e-17, -4.2825985813177425e-18, 2.6020852139652106e-17, 3.0330555775281987e-17, -3.469446951953614e-17, -1.8539857149502126e-17, -1.1926223897340549e-18, -7.28583859910259e-17, 4.85722573273506e-17, -3.469446951953614e-17, -2.6454533008646308e-17, -6.938893903907228e-18, 3.7947076036992655e-19], [0.023570226039551587, 0.016666666666666663, 0.009622504486493759, -0.005832118435198046, 0.013468700594029465, -6.329030181884132e-18, -0.005050762722761046, 0.00486009869599836, -1.3986208025063007e-17, -3.63207727782644e-18, 1.1221492485224971e-17, -3.313592889658823e-18, -6.5052130349130266e-18, -1.1370570283941728e-17, -1.0130514049161432e-18, 3.5236570605778894e-18, 6.396792817664476e-18, -3.2526065174565133e-19, -8.61940727125976e-18, -4.784042086092288e-18, 7.064254780100865e-19], [0.007856742013183862, 0.003174603174603174, 0.005498573992282148, -0.003888078956798696, 0.005892556509887892, 0.001086751106161041, -0.0007482611441127474, -1.853308088592409e-18, 0.0016731627831578666, -1.9208589661359396e-18, 0.0008365813915789378, -0.0010867511061610447, 0.000654728501098653, -6.194987217981139e-18, -1.4171919998748513e-19, 1.6466320494623599e-18, 1.7110065534536867e-18, 7.589415207398531e-19, -2.6664597179565375e-18, -3.1564682779431502e-18, 6.606327592991509e-19], [0.07071067811865475, 0.1, -8.673617379884035e-19, 0.052489065916782374, 6.938893903907228e-18, -6.7220534694101275e-18, 0.010101525445522123, 1.214306433183765e-17, -1.9081958235744878e-17, -2.439454888092385e-18, 0.0, 2.461138931542095e-17, -2.0816681711721685e-17, -8.782037597132586e-18, -8.131516293641283e-19, -8.933825901280557e-17, 4.336808689942018e-17, -2.6020852139652106e-17, -1.5395670849294163e-17, -4.119968255444917e-18, -3.2526065174565133e-19], [0.011785113019775792, 0.011904761904761906, 0.0041239304942116105, -5.421010862427522e-19, 0.007576144084141576, -2.425902360936316e-18, -0.003928371006591925, 0.004860098695998363, -4.933119884809045e-18, -1.3662641439211864e-18, -0.0012548720873683963, 0.0010867511061610428, -2.3310346708438345e-18, -4.6010829694853594e-18, -6.2849844686269085e-19, 1.6805133673525319e-18, 3.577867169202165e-18, 0.0, -4.580754178751256e-18, -1.938011383317839e-18, -1.2705494208814505e-20], [0.04714045207910317, 0.0761904761904762, 8.673617379884035e-19, 0.052489065916782374, 5.204170427930421e-18, -3.0357660829594124e-18, 0.01795826745870597, 8.673617379884035e-18, -1.0408340855860843e-17, -1.179069862577986e-18, 0.00250974417473679, 2.3215479018345864e-17, -1.3877787807814457e-17, -4.716279450311944e-18, -3.2526065174565133e-19, -1.0755285551056204e-16, 3.122502256758253e-17, -2.0816681711721685e-17, -7.697835424647081e-18, -1.0842021724855044e-18, -1.0842021724855044e-19], [0.010101525445522107, 0.010714285714285713, 0.0041239304942116105, 0.0006480131594664499, 0.007576144084141574, 0.00036225036872034576, -0.003367175148507363, 0.004860098695998364, 0.0007529232524210378, -1.4107333736520372e-18, -0.0013689513680382515, 0.0010867511061610413, 0.0005356869554443509, -4.125050453128443e-18, -5.768294370801785e-19, -0.00010413965894473626, 5.583641188300348e-18, 0.00013040347967028243, -4.458781434346637e-18, -1.782157321023048e-18, -8.046812998915853e-21], [0.006734350297014739, 0.0029761904761904756, 0.005154913117764514, -0.0032400657973322465, 0.005611958580845611, 0.0014490014748813893, -0.0008417937871268414, 9.720197391996534e-05, 0.002133282548526282, 0.00014847846772912289, 0.0006844756840191313, -0.0010274737730977149, 0.0007142492739258036, 0.00024146032511805312, -2.439984283684419e-19, 5.206982947236986e-05, -2.525252525252341e-05, -6.520173983514082e-05, 9.91899501072666e-05, -2.6668832344301646e-18, 5.834998215398061e-19], [0.010101525445522109, -0.0017857142857142867, 0.011340808859081933, -0.0038880789567986946, 0.0036477730775496455, 0.005433755530805218, 0.002525381361380526, -0.003985280930718661, 0.0033881546358946846, 0.0013363062095621226, -0.0004106854104114773, 0.0008101235518655045, -0.0011479006188093303, 0.0010865714630312556, 0.0001368951368038301, -3.124189768342139e-05, 2.5252525252526364e-05, 2.794360278649408e-05, -9.918995010727389e-05, 0.0001249675907336826, 5.1977630869055855e-18]], dtype=np.float64),
    [[np.array([[0.0, 0.0]], dtype=np.float64), np.array([[1.0, 0.0]], dtype=np.float64), np.array([[0.0, 1.0]], dtype=np.float64)], [np.array([[0.75, 0.25], [0.5, 0.5], [0.25, 0.75]], dtype=np.float64), np.array([[0.0, 0.25], [0.0, 0.5], [0.0, 0.75]], dtype=np.float64), np.array([[0.25, 0.0], [0.5, 0.0], [0.75, 0.0]], dtype=np.float64)], [np.array([[0.16666666666666666, 0.16666666666666666], [0.16666666666666666, 0.4166666666666667], [0.16666666666666666, 0.6666666666666666], [0.4166666666666667, 0.16666666666666666], [0.4166666666666667, 0.4166666666666667], [0.6666666666666666, 0.16666666666666666]], dtype=np.float64)], []],
    [[np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64), np.array([[[[1.0]]]], dtype=np.float64)], [np.array([[[[1.0], [0.0], [0.0]]], [[[0.0], [1.0], [0.0]]], [[[0.0], [0.0], [1.0]]]], dtype=np.float64), np.array([[[[1.0], [0.0], [0.0]]], [[[0.0], [1.0], [0.0]]], [[[0.0], [0.0], [1.0]]]], dtype=np.float64), np.array([[[[1.0], [0.0], [0.0]]], [[[0.0], [1.0], [0.0]]], [[[0.0], [0.0], [1.0]]]], dtype=np.float64)], [np.array([[[[1.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [1.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [1.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [1.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [1.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [1.0]]]], dtype=np.float64)], []],
    0,
    basix.MapType.identity,
    basix.SobolevSpace.L2,
    False,
    -1,
    5,
    basix.PolysetType.standard, dtype=np.float64
)

# Create conforming Crouzeix-Raviart degree 5 on a triangle
e = basix.ufl.custom_element(
    basix.CellType.triangle,
    (),
    np.array([[0.7071067811865476, 3.8163916471489756e-17, 3.642919299551295e-17, -9.280770596475918e-17, 1.734723475976807e-17, -3.729655473350135e-17, 9.020562075079397e-17, 1.3877787807814457e-17, -1.97758476261356e-16, -4.629543276513104e-17, -1.1796119636642288e-16, -1.3877787807814457e-17, -8.326672684688674e-17, -4.336808689942018e-18, -2.494003809895562e-16, -8.023096076392733e-17, -4.85722573273506e-17, -8.326672684688674e-17, -7.45931094670027e-17, -1.7211709488207383e-16, -4.0657581468206416e-17, -2.636779683484747e-16, -2.7755575615628914e-17, -2.5326962749261384e-16, -2.7755575615628914e-17, -1.1340754724198376e-16, -4.618701254788249e-17, -3.289604916592581e-16], [0.23570226039551587, -0.08333333333333331, 0.1443375672974064, -3.241764495731658e-17, 1.2793585635328952e-17, -1.905485318143274e-17, 4.765068548073792e-17, -3.67544536472586e-17, -4.9764879717084654e-17, -7.36478206978669e-17, -2.1521413123837263e-17, 4.732542482899227e-17, -1.474514954580286e-17, -3.164515090942066e-17, -1.2481697516238078e-16, 1.2115959277525512e-17, -7.079840186330344e-17, 3.686287386450715e-18, -5.800481622797449e-17, -6.356219939490995e-17, -1.403647943048255e-16, -9.562663161322149e-17, 5.453536927602087e-17, -2.4340338772299575e-17, -3.778444571111983e-17, -5.9834407394043776e-18, -1.2259616065379841e-16, -1.1222826562116896e-16], [0.11785113019775793, -0.06666666666666665, 0.11547005383792512, 0.013608276348795408, -0.02357022603955157, 0.030429030972509204, 2.532797918879809e-17, -4.079988300334514e-17, -1.7718235190665454e-17, -6.132386189223126e-17, -1.936994943781134e-17, 5.23347776790542e-17, -2.2836008257975937e-18, -9.859887022513683e-18, -9.028130777684257e-17, 3.175420640137965e-17, -6.200958800259282e-17, 3.503328269843786e-18, -2.5388965561000398e-17, -5.178275042546081e-17, -1.0301157439064181e-16, -5.871293577187908e-17, 6.842332147920238e-17, -1.0740377771184528e-18, -1.3237430899690206e-17, 1.7829725902347802e-17, -8.268555636595188e-17, -8.117514441976476e-17], [0.07071067811865475, -0.049999999999999996, 0.08660254037844385, 0.017496355305594114, -0.030304576336566302, 0.03912303982179756, -0.002525381361380511, 0.004374088826398495, -0.005646924393157827, 0.006681531047810556, -1.8986719968737973e-17, 4.3353369701961634e-17, -6.265026254807229e-18, 6.016117682322672e-19, -7.814880561084903e-17, 2.9947246896869814e-17, -5.69193435060681e-17, 8.316804750853161e-19, -8.280091166545608e-18, -4.054852638983773e-17, -8.980425513999522e-17, -4.1146742995245775e-17, 6.270309622815728e-17, -1.9025948182107687e-18, -2.667465569581402e-18, 2.6297352148648982e-17, -5.4706079502117117e-17, -8.057239744953934e-17], [0.047140452079103175, -0.03809523809523809, 0.0659828879073858, 0.01749635530559412, -0.03030457633656631, 0.039123039821797566, -0.0044895668646764775, 0.007776157913597363, -0.010038976698947235, 0.011878277418329928, 0.0005019488349473441, -0.0008694008849287948, 0.0011223917161691233, -0.0013280317881493183, 0.001505846504842015, 2.5018351113256347e-17, -4.890311633748201e-17, -1.8994730385789936e-18, -9.367323301614894e-18, -2.944531535553386e-17, -7.960270985732529e-17, -3.3108011550914565e-17, 5.505134468979675e-17, -6.634734629587805e-18, 5.503249986108157e-18, 2.2750869039067692e-17, -3.3367361158982284e-17, -7.078629540618565e-17], [0.03367175148507369, -0.029761904761904753, 0.05154913117764515, 0.016200328986661218, -0.028059792904228064, 0.03622503687203477, -0.005611958580845603, 0.00972019739199671, -0.012548720873684041, 0.014847846772912425, 0.001140792806698535, -0.0019759111021109502, 0.0025508902640207333, -0.00301825406397573, 0.0034223784200955966, -0.00010413965894471746, 0.00018037518037513714, -0.00023286335655408054, 0.0002755276391868479, -0.0003124189768342348, 0.0003453921745152132, -2.3421309058504424e-17, 5.089995843447645e-17, -5.499786193142485e-18, 1.595744425505738e-17, 1.9284308801241252e-17, -1.9306329181174787e-17, -6.521078594452839e-17], [0.23570226039551587, 0.16666666666666669, 4.336808689942018e-18, -1.9081958235744878e-17, 0.0, -3.599551212651875e-17, -2.42861286636753e-17, 3.469446951953614e-18, -8.847089727481716e-17, -3.5236570605778894e-18, -6.591949208711867e-17, -2.0816681711721685e-17, -7.632783294297951e-17, -1.734723475976807e-17, -3.67544536472586e-17, -1.2923689896027213e-16, -3.642919299551295e-17, -8.673617379884035e-17, -2.3418766925686896e-17, -7.269575566515307e-17, -1.81603863891322e-18, -3.2959746043559335e-17, -1.734723475976807e-17, -1.9255430583342559e-16, -2.7755575615628914e-17, -1.4972832002024816e-16, -1.1167282376600696e-17, -5.0415401020575956e-17], [0.058925565098878974, 0.016666666666666666, 0.02886751345948129, -0.02041241452319315, 0.023570226039551577, -9.574860435762611e-18, 6.179952383167375e-18, -6.179952383167375e-18, -2.0491421059976034e-17, -1.0279591847878189e-17, 6.505213034913027e-19, -3.7947076036992655e-19, -1.2793585635328952e-17, -2.507217523872729e-17, -1.599589957154224e-17, -1.7137170588849004e-17, -9.269928574751063e-18, -1.5178830414797062e-18, -2.4638494369733088e-17, -2.5608347094339262e-17, -1.8501740666875682e-17, -6.5052130349130266e-18, -8.131516293641283e-18, -2.4990860075790877e-17, -2.667137344314341e-17, -3.1553671351117196e-17, -3.367125371925295e-17, -1.8670406102497695e-17], [0.023570226039551587, -1.4179331537036988e-18, 0.019245008972987525, -0.009720197391996737, 0.010101525445522102, 0.00434700442464417, 0.0033671751485073714, -0.00486009869599837, 0.003764616262105203, -8.643600649815711e-18, 1.3247595295057257e-18, 1.455202603382888e-18, -4.687480330105298e-18, -1.2195156858093789e-17, -1.1262244365283008e-17, -4.2785751748182846e-18, -1.1358711822680168e-18, 1.9668105035244854e-18, -1.1690748738003853e-17, -1.5064388322689126e-17, -1.477059362144499e-17, -3.499940138054769e-18, -1.2095630486791409e-18, -2.630884334171857e-18, -9.83066438583341e-18, -8.602572491433081e-18, -2.2262672831803183e-17, -1.1819206578410888e-17], [0.011785113019775794, -0.002380952380952382, 0.012371791482634837, -0.004374088826398532, 0.003788072042070787, 0.004890379977724695, 0.003086577219465089, -0.004374088826398533, 0.0031371802184210055, 0.0007423923386456167, -0.0006274360436842016, 0.0009780759955449409, -0.0009820927516479836, 0.0006640158940746564, -1.0241610454185533e-17, -7.618962098878925e-19, -5.618740116052198e-19, 3.417777942171102e-19, -3.388343547254015e-18, -1.234354561536029e-17, -1.1478167146288058e-17, -2.4282052317616638e-18, 4.2632227026493004e-19, 4.625329287600514e-19, -3.854820473174719e-18, -2.8856443823096136e-19, -1.5714522505653753e-17, -1.0026519351587617e-17], [0.006734350297014739, -0.002380952380952382, 0.008247860988423226, -0.001944039478399347, 0.0011223917161691215, 0.004347004424644175, 0.002244783432338246, -0.0031104631654389567, 0.002007795339789444, 0.0011878277418329928, -0.0008670025330908977, 0.0013436195494354738, -0.0013264629372907813, 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    0,
    basix.MapType.identity,
    basix.SobolevSpace.L2,
    False,
    -1,
    6,
    basix.PolysetType.standard, dtype=np.float64
)

Symfem

"conforming Crouzeix-Raviart"
↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations.

Examples

triangle degree 1	(click to view basis functions)
triangle degree 2	(click to view basis functions)
triangle degree 3	(click to view basis functions)
triangle degree 4	(click to view basis functions)
triangle degree 5	(click to view basis functions)

References

[1] Crouzeix, Michel and Raviart, Pierre-Arnaud. Conforming and nonconforming finite element methods for solving the stationary Stokes equations, Revue Française d'Automatique, Informatique et Recherche Opérationnelle 3, 33–75, 1973. [DOI: 10.1051/m2an/197307R300331] [BibTeX]

DefElement stats

Element added	04 July 2021
Element last updated	04 June 2025