Mardal–Tai–Winther

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Abbreviated names	MTW
Degrees	\(k=1\)
Polynomial subdegree	\(1\)
Polynomial superdegree	\(d-1\)
Reference cells	triangle, tetrahedron
Finite dimensional space	\(\mathcal{Z}^{(19)}_{k}\) (triangle) \(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(20)}_{k}\) (tetrahedron) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{Z}^{(19)}_k=\left\{\boldsymbol{p}\in\mathcal{P}_{k}^d\middle\|\operatorname{div}\boldsymbol{p}\text{ is constant, and }\boldsymbol{p}\cdot\boldsymbol{n}_{e_i}\text{ is linear on each edge }e_i\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(20)}_k=\left\{\nabla\times(xyz(1-x-y-z)\boldsymbol{p})\middle\|\boldsymbol{p}\in\mathcal{P}_{k}^d\right\}\)
DOFs	On each facet (triangle): normal integral moments with a degree \(1\) Lagrange space, and tangent integral moments with a degree \(0\) Lagrange space On each facet (tetrahedron): normal integral moments with a degree \(1\) Lagrange space, and integral moments with a degree \(1\) Nédélec (first kind) space
Number of DOFs	triangle: \(9\) tetrahedron: \(24\)
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, 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 Mardal-Tai-Winther degree 1 on a triangle e = basix.ufl.custom_element( basix.CellType.triangle, (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], [0.23570226039551628, -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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [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, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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, 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, 0.23570226039551628, -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], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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], [0.2357022603955162, -0.03333333333333354, 0.17320508075688879, -0.02721655269758997, 0.02357022603955232, 0.030429030972510533, -1.2082349010178461e-15, 9.341485918135106e-16, 2.682316174729138e-15, 1.3085236019727553e-15, -0.23570226039551617, -0.1666666666666677, -0.05773502691896291, -8.517492267046123e-16, -0.04714045207910385, -1.35828848168984e-15, -1.457167719820518e-16, -4.440892098500626e-16, -3.2612801348363973e-15, -7.064661355915547e-16], [0.23570226039551637, -0.03333333333333367, 0.17320508075688879, -0.007776157913596246, 0.02357022603955233, 0.021735022123221936, -0.01262690680690406, 0.01020620726159774, 0.005646924393160608, -0.006681531047809442, -0.23570226039551595, -0.16666666666666757, -0.057735026918963026, -9.575673587391975e-16, -0.06397632782164099, 0.013041013273931298, -1.9081958235744878e-16, -0.014580296087995825, 0.011293848786312585, -7.095019016745141e-16], [0.2357022603955163, -0.016666666666666777, 0.18282758524338266, -0.023328473740791313, 0.03703892663358205, 0.03912303982179898, -0.008417937871269607, 0.0048600986959994865, 0.007529232524213361, 1.3984039620718036e-15, -0.23570226039551614, -0.16666666666666768, -0.0769800358919507, -8.634586101674557e-16, -0.0740778532671633, -1.366962099069724e-15, -1.478851763270228e-16, -0.009720197391997493, -3.301178774783864e-15, -8.879615792656281e-16]], 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([[0.8872983346207417, 0.1127016653792583], [0.5, 0.5], [0.1127016653792583, 0.8872983346207417], [0.7886751345948129, 0.21132486540518713], [0.21132486540518713, 0.7886751345948129]], dtype=np.float64), np.array([[0.0, 0.1127016653792583], [0.0, 0.5], [0.0, 0.8872983346207417], [0.0, 0.21132486540518713], [0.0, 0.7886751345948129]], dtype=np.float64), np.array([[0.1127016653792583, 0.0], [0.5, 0.0], [0.8872983346207417, 0.0], [0.21132486540518713, 0.0], [0.7886751345948129, 0.0]], 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.array([[[[-0.1742818525960713], [-0.15713484026367722], [-0.022136697733525185], [0.0], [0.0]], [[-0.1742818525960713], [-0.15713484026367722], [-0.022136697733525185], [0.0], [0.0]]], [[[-0.022136697733525185], [-0.15713484026367722], [-0.1742818525960713], [0.0], [0.0]], [[-0.022136697733525185], [-0.15713484026367722], [-0.1742818525960713], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [-0.3535533905932738], [-0.3535533905932738]], [[0.0], [0.0], [0.0], [0.3535533905932738], [0.3535533905932738]]]], 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]]], [[[-0.03130601816090507], [-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], [0.5], [0.5]]]], dtype=np.float64), np.array([[[[0.0], [0.0], [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]], [[0.03130601816090507], [0.2222222222222222], [0.24647175961687262], [0.0], [0.0]]], [[[0.0], [0.0], [0.0], [0.5], [0.5]], [[0.0], [0.0], [0.0], [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, 3, basix.PolysetType.standard, dtype=np.float64 ) # Create Mardal-Tai-Winther degree 1 on a tetrahedron e = basix.ufl.custom_element( basix.CellType.tetrahedron, (3, ), np.array([[0.4082482904638628, 6.678685382510707e-17, 7.28583859910259e-17, 1.3704315460216776e-16, 2.177077962350893e-16, 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FIAT	`FIAT.MardalTaiWinther` ↓ 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 Mardal-Tai-Winther degree 1 element = FIAT.MardalTaiWinther(FIAT.ufc_cell("triangle"), 3) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: triangle,1 Not implemented: tetrahedron,1 Note: This implementation includes additional DOFs that are used then filtered out when mapping the element, as described in Kirby (2018). Note: This implementation uses an alternative value of degree for this element
Symfem	`"MTW"` ↓ 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 Mardal-Tai-Winther degree 1 on a triangle element = symfem.create_element("triangle", "MTW", 1) # Create Mardal-Tai-Winther degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "MTW", 1) This implementation is used to compute the examples below and verify other implementations.

Examples

triangle degree 1	(click to view basis functions)
tetrahedron degree 1	(click to view basis functions)

References

[1] Mardal, Kent-Andre, Tai, Xue-Cheng, and Winther, Ragner. A robust finite element method for Darcy–Stokes flow, SIAM Journal on Numerical Analysis 40(5), 1605–1631, 2002. [DOI: 10.1137/S0036142901383910] [BibTeX]
[2] Tai, Xue-Cheng and Winther, Ragner. A discrete de Rham complex with enhanced smoothness, Calcolo 43, 287–306, 2006. [DOI: 10.1007/s10092-006-0124-6] [BibTeX]
[3] 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	09 January 2021
Element last updated	17 November 2025