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Lagrange

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Alternative names	Polynomial, Galerkin, DGT (facets), Hdiv trace (facets), Q (quadrilateral and hexahedron)
Exterior calculus names	\(\mathcal{P}^-_{k}\Lambda^{0}(\Delta_d)\), \(\mathcal{P}_{k}\Lambda^{0}(\Delta_d)\), \(\mathcal{Q}^-_{k}\Lambda^{0}(\square_d)\), \(\mathcal{P}^-_{k}\Lambda^{d}(\Delta_d)\), \(\mathcal{P}_{k}\Lambda^{d}(\Delta_d)\), \(\mathcal{Q}^-_{k}\Lambda^{d}(\square_d)\)
Cockburn–fu names	\(\left[S_{2,k}^\unicode{0x25FA}\right]_{0}\), \(\left[S_{1,k}^\unicode{0x25FA}\right]_{0}\), \(\left[S_{4,k}^\square\right]_{0}\), \(\left[S_{2,k}^\unicode{0x25FA}\right]_{d}\), \(\left[S_{1,k}^\unicode{0x25FA}\right]_{d}\), \(\left[S_{4,k}^\square\right]_{d}\), \(\left[S_{3,k}^\square\right]_{d}\)
Abbreviated names	P, CG, DG
Orders	\(1\leqslant k\)
Reference elements	interval, triangle, tetrahedron, quadrilateral, hexahedron, prism, pyramid
Polynomial set	\(\mathcal{P}_{k}\) (interval, triangle, tetrahedron) \(\mathcal{Q}_{k}\) (quadrilateral, hexahedron) \(\mathcal{Z}^{(14)}_{k}\) (prism) \(\mathcal{Z}^{(15)}_{k} \oplus \mathcal{Z}^{(16)}_{k}\) (pyramid) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)\leqslant k\right\}\) \(\mathcal{Z}^{(14)}_k=\operatorname{span}\left\{x_1^{p_1}x_2^{p_2}x_3^{p_3}\middle\|\max(p_1+p_2,p_3)\leqslant k\right\}\) \(\mathcal{Z}^{(15)}_k=\operatorname{span}\left\{x_1^{p_1}x_2^{p_2}x_3^{p_3}\middle\|p_3\leqslant k-1,p_1+p_3\leqslant k,p_2+p_3\leqslant k\right\}\) \(\mathcal{Z}^{(16)}_k=\operatorname{span}\left\{x_3^k\right\}\)
DOFs	On each vertex: point evaluations On each edge: point evaluations On each face: point evaluations On each volume: point evaluations
Number of DOFs	interval: \(k+1\) (A000027) triangle: \((k+1)(k+2)/2\) (A000217) tetrahedron: \((k+1)(k+2)(k+3)/6\) (A000292) quadrilateral: \((k+1)^2\) (A000290) hexahedron: \((k+1)^3\) (A000578) prism: \((k+1)^2(k+2)/2\) (A002411) pyramid: \((k+1)(k+2)(2k+3)/6\) (A000330)
Number of DOFs on subentities	vertices: \(1\) (A000012) edges: \(k-1\) (A000027) faces: \((k-1)(k-2)/2\) (A000217) (triangle), \((k-1)^2\) (A000290) (quadrilateral) volumes: \((k-1)(k-2)(k-3)/6\) (A000292) (tetrahedron), \((k-1)^3\) (A000578) (hexahedron), \((k-1)^2(k-2)/2\) (A002411) (prism), \((k-1)(k-2)(2k-3)/6\) (A000330) (pyramid)
Mapping	identity
continuity	Function values are continuous.
Notes	DGT and Hdiv trace are names given to this element when it is defined on the facets of a mesh.
Categories	Scalar-valued elements

Implementations

Basix	`basix.ElementFamily.P, ..., basix.LagrangeVariant.equispaced` ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before trying this example, you must install Basix: pip3 install git+https://github.com/fenics/basix.git This element can then be created with the following lines of Python: import basix # Create Lagrange order 1 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 2 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 3 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 1 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 2 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 3 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 1 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 2 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 3 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 1 on a hexahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.hexahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 2 on a hexahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.hexahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 1 on a prism element = basix.create_element(basix.ElementFamily.P, basix.CellType.prism, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 2 on a prism element = basix.create_element(basix.ElementFamily.P, basix.CellType.prism, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 1 on a pyramid element = basix.create_element(basix.ElementFamily.P, basix.CellType.pyramid, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange order 2 on a pyramid element = basix.create_element(basix.ElementFamily.P, basix.CellType.pyramid, 2, lagrange_variant=basix.LagrangeVariant.equispaced)
Bempp	`"P"` (triangle) ↓ Show Bempp examples ↓↑ Hide Bempp examples ↑ Before trying this example, you must install Bempp: pip3 install bempp-cl This element can then be created with the following lines of Python: import bempp.api grid = bempp.api.shapes.regular_sphere(1) # Create Lagrange order 1 element = bempp.api.function_space(grid, "P", 1)
Symfem	`"Lagrange"` (interval, triangle, tetrahedron, prism, pyramid) `"Q"` (quadrilateral, hexahedron) ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before trying this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Lagrange order 1 on a interval element = symfem.create_element("interval", "Lagrange", 1) # Create Lagrange order 2 on a interval element = symfem.create_element("interval", "Lagrange", 2) # Create Lagrange order 3 on a interval element = symfem.create_element("interval", "Lagrange", 3) # Create Lagrange order 1 on a triangle element = symfem.create_element("triangle", "Lagrange", 1) # Create Lagrange order 2 on a triangle element = symfem.create_element("triangle", "Lagrange", 2) # Create Lagrange order 3 on a triangle element = symfem.create_element("triangle", "Lagrange", 3) # Create Lagrange order 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 1) # Create Lagrange order 2 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 2) # Create Lagrange order 3 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 3) # Create Lagrange order 1 on a tetrahedron element = symfem.create_element("tetrahedron", "Lagrange", 1) # Create Lagrange order 2 on a tetrahedron element = symfem.create_element("tetrahedron", "Lagrange", 2) # Create Lagrange order 1 on a hexahedron element = symfem.create_element("hexahedron", "Q", 1) # Create Lagrange order 2 on a hexahedron element = symfem.create_element("hexahedron", "Q", 2) # Create Lagrange order 1 on a prism element = symfem.create_element("prism", "Lagrange", 1) # Create Lagrange order 2 on a prism element = symfem.create_element("prism", "Lagrange", 2) # Create Lagrange order 1 on a pyramid element = symfem.create_element("pyramid", "Lagrange", 1) # Create Lagrange order 2 on a pyramid element = symfem.create_element("pyramid", "Lagrange", 2)
UFL	`"Lagrange"` (interval, triangle, tetrahedron) `"Q"` (quadrilateral, hexahedron) ↓ Show UFL examples ↓↑ Hide UFL examples ↑ Before trying this example, you must install UFL: pip3 install UFL This element can then be created with the following lines of Python: import ufl # Create Lagrange order 1 on a interval element = ufl.FiniteElement("Lagrange", "interval", 1) # Create Lagrange order 2 on a interval element = ufl.FiniteElement("Lagrange", "interval", 2) # Create Lagrange order 3 on a interval element = ufl.FiniteElement("Lagrange", "interval", 3) # Create Lagrange order 1 on a triangle element = ufl.FiniteElement("Lagrange", "triangle", 1) # Create Lagrange order 2 on a triangle element = ufl.FiniteElement("Lagrange", "triangle", 2) # Create Lagrange order 3 on a triangle element = ufl.FiniteElement("Lagrange", "triangle", 3) # Create Lagrange order 1 on a quadrilateral element = ufl.FiniteElement("Q", "quadrilateral", 1) # Create Lagrange order 2 on a quadrilateral element = ufl.FiniteElement("Q", "quadrilateral", 2) # Create Lagrange order 3 on a quadrilateral element = ufl.FiniteElement("Q", "quadrilateral", 3) # Create Lagrange order 1 on a tetrahedron element = ufl.FiniteElement("Lagrange", "tetrahedron", 1) # Create Lagrange order 2 on a tetrahedron element = ufl.FiniteElement("Lagrange", "tetrahedron", 2) # Create Lagrange order 1 on a hexahedron element = ufl.FiniteElement("Q", "hexahedron", 1) # Create Lagrange order 2 on a hexahedron element = ufl.FiniteElement("Q", "hexahedron", 2)

Examples

interval order 1	(click to view basis functions)
interval order 2	(click to view basis functions)
interval order 3	(click to view basis functions)
triangle order 1	(click to view basis functions)
triangle order 2	(click to view basis functions)
triangle order 3	(click to view basis functions)
quadrilateral order 1	(click to view basis functions)
quadrilateral order 2	(click to view basis functions)
quadrilateral order 3	(click to view basis functions)
tetrahedron order 1	(click to view basis functions)
tetrahedron order 2	(click to view basis functions)
hexahedron order 1	(click to view basis functions)
hexahedron order 2	(click to view basis functions)
prism order 1	(click to view basis functions)
prism order 2	(click to view basis functions)
pyramid order 1	(click to view basis functions)
pyramid order 2	(click to view basis functions)

References

Arnold, Douglas N. and Logg, Anders. Periodic table of the finite elements, SIAM News 47, 2014. [sinews.siam.org/Details-Page/periodic-table-of-the-finite-elements] [BibTeX]
Cockburn, Bernardo and Fu, Guosheng. A systematic construction of finite element commuting exact sequences, SIAM journal on numerical analysis 55, 1650–1688, 2017. [DOI: 10.1137/16M1073352] [BibTeX]

DefElement stats

Element added	30 December 2020
Element last updated	02 August 2022