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Taylor

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Alternative names	discontinuous Taylor
Degrees	\(0\leqslant k\) where \(k\) is the Polynomial superdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k\)
Reference elements	interval, triangle, tetrahedron
Polynomial set	\(\mathcal{P}_{k}\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On the interior of the reference element: integral over cell, and point evaluations at midpoint of derivatives up to order \(k\)
Number of DOFs	interval: \(k+1\) (A000027) triangle: \((k+1)(k+2)/2\) (A000217) tetrahedron: \((k+1)(k+2)(k+3)/6\) (A000292)
Mapping	identity
continuity	Function values are continuous.
Categories	Scalar-valued elements

Implementations

This element is implemented in FIAT , Symfem , and (legacy) UFL.↓ Show implementation detail ↓

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FIAT	`FIAT.DiscontinuousTaylor` ↓ 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 Taylor degree 1 element = FIAT.DiscontinuousTaylor(FIAT.ufc_cell("interval"), 1) # Create Taylor degree 2 element = FIAT.DiscontinuousTaylor(FIAT.ufc_cell("interval"), 2) # Create Taylor degree 3 element = FIAT.DiscontinuousTaylor(FIAT.ufc_cell("interval"), 3) # Create Taylor degree 1 element = FIAT.DiscontinuousTaylor(FIAT.ufc_cell("triangle"), 1) # Create Taylor degree 2 element = FIAT.DiscontinuousTaylor(FIAT.ufc_cell("triangle"), 2) # Create Taylor degree 3 element = FIAT.DiscontinuousTaylor(FIAT.ufc_cell("triangle"), 3) This implementation is correct for all the examples below.
Symfem	`"Taylor"` ↓ 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 Taylor degree 1 on a interval element = symfem.create_element("interval", "Taylor", 1) # Create Taylor degree 2 on a interval element = symfem.create_element("interval", "Taylor", 2) # Create Taylor degree 3 on a interval element = symfem.create_element("interval", "Taylor", 3) # Create Taylor degree 1 on a triangle element = symfem.create_element("triangle", "Taylor", 1) # Create Taylor degree 2 on a triangle element = symfem.create_element("triangle", "Taylor", 2) # Create Taylor degree 3 on a triangle element = symfem.create_element("triangle", "Taylor", 3) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"TDG"` ↓ 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 Taylor degree 1 on a interval element = ufl_legacy.FiniteElement("TDG", "interval", 1) # Create Taylor degree 2 on a interval element = ufl_legacy.FiniteElement("TDG", "interval", 2) # Create Taylor degree 3 on a interval element = ufl_legacy.FiniteElement("TDG", "interval", 3) # Create Taylor degree 1 on a triangle element = ufl_legacy.FiniteElement("TDG", "triangle", 1) # Create Taylor degree 2 on a triangle element = ufl_legacy.FiniteElement("TDG", "triangle", 2) # Create Taylor degree 3 on a triangle element = ufl_legacy.FiniteElement("TDG", "triangle", 3)

Examples

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

DefElement stats

Element added	01 March 2021
Element last updated	27 September 2024