an encyclopedia of finite element definitions

Taylor

Click here to read what the information on this page means.

Alternative names	discontinuous Taylor
Degrees	\(0\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k\)
Reference cells	interval, triangle, tetrahedron
Finite dimensional space	\(\mathcal{P}_{k}\) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On the interior of the reference cell: 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 ↓

↑ Hide implementation detail ↑

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	04 June 2025