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Arnold–Winther

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Alternative names	conforming Arnold–Winther
Degrees	\(2\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Reference cells	triangle
Finite dimensional space	\(\mathcal{Z}^{(2)}_{k} \oplus \mathcal{Z}^{(3)}_{k+1}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{Z}^{(2)}_k=\left\{\mathbf{M}\in\mathcal{P}_{k}^{d\times d}\middle\|\mathbf{M}^t=\mathbf{M}\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(3)}_k=\left\{\mathbf{M}\in\mathcal{P}_{k}^{d\times d}\middle\|\mathbf{M}^t=\mathbf{M}\text{ and }\operatorname{div}\mathbf{M}=0\right\}\)
DOFs	On each vertex: point evaluations of three components On each edge: integral moments of normal-normal and normal-tangent inner products with a degree \(k-1\) Lagrange space On each face: integral moments of three components with a degree \(k-2\) Lagrange space, and integral moments of tensor dot product with \(\frac{\partial}{\partial(x, y)}x^2y^2(1-x-y)^2f\) for each degree \(k-3\) polynomial \(f\) in a degree \(k-3\) Lagrange space
Number of DOFs	triangle: \((3k^2+11k+14)/2\)
Mapping	double contravariant Piola
continuity	Inner products with normals to facets are continuous
Categories	Matrix-valued elements

Implementations

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

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Basix.UFL	Uses custom element code ↓ Show Basix.UFL examples ↓↑ Hide 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 Arnold-Winther degree 2 on a triangle e = basix.ufl.custom_element( basix.CellType.triangle, (2, 2), 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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865489, -3.608224830031759e-16, -5.967448757360216e-16, 3.434752482434078e-15, -4.926614671774132e-16, 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-0.010101525445521904, 0.014580296087994955, -0.01129384878631672, -6.24188743227061e-16], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.07071067811865506, -0.05000000000000034, 0.08660254037844436, 0.01749635530559457, -0.030304576336566497, 0.03912303982179802, -0.0025253813613811296, 0.0043740888263988946, -0.005646924393157328, 0.006681531047811082]], 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([[0.8872983346207417, 0.1127016653792583], [0.5, 0.5], [0.1127016653792583, 0.8872983346207417]], dtype=np.float64), np.array([[0.0, 0.1127016653792583], [0.0, 0.5], [0.0, 0.8872983346207417]], dtype=np.float64), np.array([[0.1127016653792583, 0.0], [0.5, 0.0], [0.8872983346207417, 0.0]], dtype=np.float64)], [np.array([[0.659027622374092, 0.231933368553031], [0.659027622374092, 0.109039009072877], [0.231933368553031, 0.659027622374092], [0.231933368553031, 0.109039009072877], [0.109039009072877, 0.659027622374092], [0.109039009072877, 0.231933368553031]], dtype=np.float64)], []], [[np.array([[[[1.0]], [[0.0]], [[0.0]], [[0.0]]], [[[0.0]], [[1.0]], [[0.0]], [[0.0]]], [[[0.0]], [[0.0]], [[0.0]], [[1.0]]]], dtype=np.float64), np.array([[[[1.0]], [[0.0]], [[0.0]], [[0.0]]], [[[0.0]], [[1.0]], [[0.0]], [[0.0]]], [[[0.0]], [[0.0]], [[0.0]], [[1.0]]]], dtype=np.float64), np.array([[[[1.0]], [[0.0]], [[0.0]], [[0.0]]], [[[0.0]], [[1.0]], [[0.0]], [[0.0]]], [[[0.0]], [[0.0]], [[0.0]], [[1.0]]]], dtype=np.float64)], [np.array([[[[0.12323587980843631], [0.1111111111111111], [0.015653009080452536]], [[0.12323587980843631], [0.1111111111111111], [0.015653009080452536]], [[0.12323587980843631], [0.1111111111111111], [0.015653009080452536]], [[0.12323587980843631], [0.1111111111111111], [0.015653009080452536]]], [[[0.12323587980843631], [0.1111111111111111], [0.015653009080452536]], [[0.12323587980843631], [0.1111111111111111], [0.015653009080452536]], [[-0.12323587980843631], [-0.1111111111111111], [-0.015653009080452536]], [[-0.12323587980843631], [-0.1111111111111111], [-0.015653009080452536]]], [[[0.015653009080452536], [0.1111111111111111], [0.12323587980843631]], [[0.015653009080452536], [0.1111111111111111], [0.12323587980843631]], [[0.015653009080452536], [0.1111111111111111], [0.12323587980843631]], [[0.015653009080452536], [0.1111111111111111], [0.12323587980843631]]], [[[0.015653009080452536], [0.1111111111111111], [0.12323587980843631]], [[0.015653009080452536], [0.1111111111111111], [0.12323587980843631]], [[-0.015653009080452536], [-0.1111111111111111], [-0.12323587980843631]], [[-0.015653009080452536], [-0.1111111111111111], [-0.12323587980843631]]]], dtype=np.float64), np.array([[[[0.24647175961687262], [0.2222222222222222], [0.03130601816090507]], [[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]], [[-0.24647175961687262], [-0.2222222222222222], [-0.03130601816090507]], [[0.0], [0.0], [0.0]]], [[[0.03130601816090507], [0.2222222222222222], [0.24647175961687262]], [[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]], [[-0.03130601816090507], [-0.2222222222222222], [-0.24647175961687262]], [[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], [0.0]], [[0.24647175961687262], [0.2222222222222222], [0.03130601816090507]]], [[[0.0], [0.0], [0.0]], [[0.24647175961687262], [0.2222222222222222], [0.03130601816090507]], [[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]], [[0.03130601816090507], [0.2222222222222222], [0.24647175961687262]]], [[[0.0], [0.0], [0.0]], [[0.03130601816090507], [0.2222222222222222], [0.24647175961687262]], [[0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0]]]], dtype=np.float64)], [np.array([[[[0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.0], [0.0], [0.0], [0.0], [0.0], [0.0]], [[0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333], [0.08333333333333333]]]], dtype=np.float64)], []], 0, basix.MapType.doubleContravariantPiola, basix.SobolevSpace.HDivDiv, False, -1, 3, basix.PolysetType.standard, dtype=np.float64 ) # Create Arnold-Winther degree 3 on a triangle e = basix.ufl.custom_element( basix.CellType.triangle, (2, 2), 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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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, 0.7071067811865476, 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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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, 0.23570226039551584, -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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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, 0.11785113019775792, -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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.11785113019775792, -0.06666666666666662, 0.1154700538379251, 0.01360827634879535, -0.023570226039551542, 0.030429030972509173, 1.0408340855860843e-16, -7.632783294297951e-17, 4.163336342344337e-17, -1.0408340855860843e-16, 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basix.PolysetType.standard, dtype=np.float64 )
FIAT	`FIAT.ArnoldWinther` ↓ Show FIAT examples ↓↑ Hide FIAT examples ↑ Before running this example, you must install FIAT: pip3 install git+https://github.com/firedrakeproject/fiat.git This element can then be created with the following lines of Python: import FIAT # Create Arnold-Winther degree 2 element = FIAT.ArnoldWinther(FIAT.ufc_cell("triangle"), 3) This implementation is incorrect for this element. Note: This implementation includes additional DOFs that are used then filtered out when mapping the element, as described in Kirby (2018). Note: This element uses the Lagrange superdegree as the canonical degree of this element
Symfem	`"AW"` ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before running this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Arnold-Winther degree 2 on a triangle element = symfem.create_element("triangle", "AW", 2) # Create Arnold-Winther degree 3 on a triangle element = symfem.create_element("triangle", "AW", 3) This implementation is used to compute the examples below and verify other implementations.

Examples

triangle degree 2	(click to view basis functions)
triangle degree 3	(click to view basis functions)

References

[1] Arnold, Douglas N. and Winther, Ragnar. Mixed finite elements for elasticity, Numerische Mathematik 92(3), 401–419, 2002. [DOI: 10.1007/s002110100348] [BibTeX]
[2] Kirby, Robert C. A general approach to transforming finite elements, SMAI Journal of Computational Mathematics 4, 197–224, 2018. [DOI: 10.5802/smai-jcm.33] [BibTeX]

DefElement stats

Element added	10 February 2021
Element last updated	17 November 2025