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Vector Lagrange

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Degrees	\(0\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k\)
Reference cells	triangle, tetrahedron
Finite dimensional space	\(\mathcal{P}_{k}^d\) ↓ 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 each vertex: point evaluations in coordinate directions On each edge: point evaluations in coordinate directions On each face: point evaluations in coordinate directions On each volume: point evaluations in coordinate directions
Number of DOFs	triangle: \((k+1)(k+2)\) (A002378) tetrahedron: \((k+1)(k+2)(k+3)/2\) (A027480)
Mapping	identity
continuity	Function values are continuous.
Categories	Vector-valued elements

Implementations

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

↑ Hide implementation detail ↑

Basix.UFL	`basix.ElementFamily.P, lagrange_variant=basix.LagrangeVariant.equispaced, shape=(dim,)` ↓ Show Basix.UFL examples ↓↑ Hide Basix.UFL examples ↑ Before running this example, you must install Basix.UFL: pip3 install fenics-basix fenics-ufl This element can then be created with the following lines of Python: import basix import basix.ufl # Create vector Lagrange degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced, shape=(2,)) # Create vector Lagrange degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced, shape=(2,)) # Create vector Lagrange degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced, shape=(3,)) # Create vector Lagrange degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced, shape=(3,)) This implementation is correct for all the examples below.
Symfem	`"vector Lagrange"` ↓ 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 vector Lagrange degree 1 on a triangle element = symfem.create_element("triangle", "vector Lagrange", 1) # Create vector Lagrange degree 2 on a triangle element = symfem.create_element("triangle", "vector Lagrange", 2) # Create vector Lagrange degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "vector Lagrange", 1) # Create vector Lagrange degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "vector Lagrange", 2) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"Lagrange"` ↓ 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 vector Lagrange degree 1 on a triangle element = ufl_legacy.VectorElement("Lagrange", "triangle", 1) # Create vector Lagrange degree 2 on a triangle element = ufl_legacy.VectorElement("Lagrange", "triangle", 2) # Create vector Lagrange degree 1 on a tetrahedron element = ufl_legacy.VectorElement("Lagrange", "tetrahedron", 1) # Create vector Lagrange degree 2 on a tetrahedron element = ufl_legacy.VectorElement("Lagrange", "tetrahedron", 2)

Examples

triangle degree 1	(click to view basis functions)
triangle degree 2	(click to view basis functions)
tetrahedron degree 1	(click to view basis functions)
tetrahedron degree 2	(click to view basis functions)

DefElement stats

Element added	30 December 2020
Element last updated	04 June 2025