Mardal–Tai–Winther

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Abbreviated names	MTW
Degrees	\(k=3\)
Polynomial subdegree	\(1\)
Polynomial superdegree	\(k+d-2\)
Reference elements	triangle, tetrahedron
Polynomial set	\(\mathcal{Z}^{(17)}_{k}\) (triangle) \(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(18)}_{k}\) (tetrahedron) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{Z}^{(17)}_k=\left\{\boldsymbol{p}\in\mathcal{P}_{k}^d\middle\|\operatorname{div}\boldsymbol{p}\text{ is constant, and }\boldsymbol{p}\cdot\boldsymbol{n}_{e_i}\text{ is linear on each edge }e_i\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(18)}_k=\left\{\nabla\times(xyz(1-x-y-z)\boldsymbol{p})\middle\|\boldsymbol{p}\in\mathcal{P}_{k}^d\right\}\)
DOFs	On each facet (triangle): normal integral moments with an degree \(1\) Lagrange space, and tangent integral moments with an degree \(0\) Lagrange space On each facet (tetrahedron): normal integral moments with an degree \(1\) Lagrange space, and integral moments with an 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 FIAT , Symfem , and (legacy) UFL.↓ Show implementation detail ↓

Examples

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

References

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]
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]
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	27 September 2024

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 3 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,3 Not implemented: tetrahedron,3 Note: This implementation includes additional DOFs that are used then filtered out when mapping the element, as described in Kirby (2018).
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 3 on a triangle element = symfem.create_element("triangle", "MTW", 3) # Create Mardal-Tai-Winther degree 3 on a tetrahedron element = symfem.create_element("tetrahedron", "MTW", 3) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"MTW"` ↓ Show (legacy) UFL examples ↓↑ Hide (legacy) UFL examples ↑ Before running this example, you must install (legacy) UFL: pip3 install setuptools pip3 install fenics-ufl-legacy This element can then be created with the following lines of Python: import ufl_legacy # Create Mardal-Tai-Winther degree 3 on a triangle element = ufl_legacy.FiniteElement("MTW", "triangle", 3)