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Vector dPc

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Degrees	\(0\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k\)
Lagrange subdegree	\(\operatorname{floor}(k/d)\)
Lagrange superdegree	\(k\)
Reference cells	interval, quadrilateral, hexahedron
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 the interior of the reference cell: point evaluations
Number of DOFs	quadrilateral: \((k+1)(k+2)\) (A002378) hexahedron: \((k+1)(k+2)(k+3)/2\) (A027480)
Mapping	identity
continuity	Discontinuous.
Categories	Scalar-valued elements

Implementations

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

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Basix.UFL	`basix.ElementFamily.DPC, discontinuous=True, dpc_variant=basix.DPCVariant.simplex_equispaced, shape=(dim,)` ↓ 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 # Create vector dPc degree 1 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.DPC, basix.CellType.quadrilateral, 1, dpc_variant=basix.DPCVariant.simplex_equispaced, discontinuous=True, shape=(2,)) # Create vector dPc degree 2 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.DPC, basix.CellType.quadrilateral, 2, dpc_variant=basix.DPCVariant.simplex_equispaced, discontinuous=True, shape=(2,)) # Create vector dPc degree 3 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.DPC, basix.CellType.quadrilateral, 3, dpc_variant=basix.DPCVariant.simplex_equispaced, discontinuous=True, shape=(2,)) # Create vector dPc degree 1 on a hexahedron element = basix.ufl.element(basix.ElementFamily.DPC, basix.CellType.hexahedron, 1, dpc_variant=basix.DPCVariant.simplex_equispaced, discontinuous=True, shape=(3,)) # Create vector dPc degree 2 on a hexahedron element = basix.ufl.element(basix.ElementFamily.DPC, basix.CellType.hexahedron, 2, dpc_variant=basix.DPCVariant.simplex_equispaced, discontinuous=True, shape=(3,)) This implementation is correct for all the examples below.
Symfem	`"vector dPc"` ↓ 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 dPc degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "vector dPc", 1) # Create vector dPc degree 2 on a quadrilateral element = symfem.create_element("quadrilateral", "vector dPc", 2) # Create vector dPc degree 3 on a quadrilateral element = symfem.create_element("quadrilateral", "vector dPc", 3) # Create vector dPc degree 1 on a hexahedron element = symfem.create_element("hexahedron", "vector dPc", 1) # Create vector dPc degree 2 on a hexahedron element = symfem.create_element("hexahedron", "vector dPc", 2) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"DPC"` ↓ 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 dPc degree 1 on a quadrilateral element = ufl_legacy.VectorElement("DPC", "quadrilateral", 1) # Create vector dPc degree 2 on a quadrilateral element = ufl_legacy.VectorElement("DPC", "quadrilateral", 2) # Create vector dPc degree 3 on a quadrilateral element = ufl_legacy.VectorElement("DPC", "quadrilateral", 3) # Create vector dPc degree 1 on a hexahedron element = ufl_legacy.VectorElement("DPC", "hexahedron", 1) # Create vector dPc degree 2 on a hexahedron element = ufl_legacy.VectorElement("DPC", "hexahedron", 2)

Examples

quadrilateral degree 1	(click to view basis functions)
quadrilateral degree 2	(click to view basis functions)
quadrilateral degree 3	(click to view basis functions)
hexahedron degree 1	(click to view basis functions)
hexahedron degree 2	(click to view basis functions)

DefElement stats

Element added	30 March 2021
Element last updated	04 June 2025