Lagrange

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Alternative names	Polynomial, Galerkin, DGT (facets), Hdiv trace (facets), Q (quadrilateral and hexahedron), Gauss–Lobatto–Legendre (GLL variant), Lobatto (Lobatto variant)
De Rham complex families	\(\left[S_{2,k}^\unicode{0x25FA}\right]_{0}\) or \(\mathcal{P}^-_{k}\Lambda^{0}(\Delta_d)\), \(\left[S_{1,k}^\unicode{0x25FA}\right]_{0}\) or \(\mathcal{P}_{k}\Lambda^{0}(\Delta_d)\), \(\left[S_{4,k}^\square\right]_{0}\) or \(\mathcal{Q}^-_{k}\Lambda^{0}(\square_d)\), \(\left[S_{2,k}^\unicode{0x25FA}\right]_{d}\) or \(\mathcal{P}^-_{k}\Lambda^{d}(\Delta_d)\), \(\left[S_{1,k}^\unicode{0x25FA}\right]_{d}\) or \(\mathcal{P}_{k}\Lambda^{d}(\Delta_d)\), \(\left[S_{4,k}^\square\right]_{d}\) or \(\mathcal{Q}^-_{k}\Lambda^{d}(\square_d)\), \(\left[S_{3,k}^\square\right]_{d}\)
Abbreviated names	P, CG, DG, GLL (GLL variant)
Variants	equispaced: The variant has its point evaluations at equally spaced points. GLL: This variant has its point evaluations at GLL points. Lobatto: This variant uses integrals against L2 duals of Lobatto polynomials in the place of point evaluations
Degrees	\(1\leqslant k\) where \(k\) is the Lagrange superdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	interval: \(k\) triangle: \(k\) tetrahedron: \(k\) quadrilateral: \(dk\) hexahedron: \(dk\) prism: \(2k\) pyramid: undefined
Lagrange subdegree	\(k\)
Lagrange superdegree	\(k\)
Reference elements	interval, triangle, tetrahedron, quadrilateral, hexahedron, prism, pyramid
Polynomial set	\(\mathcal{P}_{k}\) (interval, triangle, tetrahedron) \(\mathcal{Q}_{k}\) (quadrilateral, hexahedron) \(\mathcal{Z}^{(15)}_{k}\) (prism) \(\mathcal{P}_{k} \oplus \mathcal{Z}^{(16)}_{k}\) (pyramid) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)\leqslant k\right\}\) \(\mathcal{Z}^{(15)}_k=\operatorname{span}\left\{x_1^{p_1}x_2^{p_2}x_3^{p_3}\middle\|\max(p_1+p_2,p_3)\leqslant k\right\}\) \(\mathcal{Z}^{(16)}_k=\operatorname{span}\left\{\frac{x_1^{p_1}x_2^{p_2}}{(1-x_3)^{\min(p_1,p_2)}}\middle\|(p_1,p_2)=(1,k)\text{ or }(p_1,p_2)=(k,1)\text{ or }p_1\leqslant k-1,p_2\leqslant k-1,k+1\leqslant p_1+p_2\right\}\)
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. For the Lobatto variant, the derivatives of most of the basis functions are orthogonal.
Categories	Scalar-valued elements

Implementations

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

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Basix	`basix.ElementFamily.P` ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before running this example, you must install Basix: pip3 install fenics-basix This element can then be created with the following lines of Python: import basix # Create Lagrange (equispaced variant) degree 1 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 3 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 3 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 3 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a hexahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.hexahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a hexahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.hexahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a prism element = basix.create_element(basix.ElementFamily.P, basix.CellType.prism, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a prism element = basix.create_element(basix.ElementFamily.P, basix.CellType.prism, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (GLL variant) degree 1 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 2 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 3 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 3, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 4 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 4, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 1 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 2 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.gll_warped) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,1,equispaced; interval,2,equispaced; interval,3,equispaced; triangle,1,equispaced; triangle,2,equispaced; triangle,3,equispaced; quadrilateral,1,equispaced; quadrilateral,2,equispaced; quadrilateral,3,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced; prism,1,equispaced; prism,2,equispaced; interval,1,gll; interval,2,gll; interval,3,gll; interval,4,gll; quadrilateral,1,gll; quadrilateral,2,gll Not implemented: pyramid,1,equispaced; pyramid,2,equispaced; interval,1,lobatto; interval,2,lobatto; interval,3,lobatto; quadrilateral,1,lobatto; quadrilateral,2,lobatto; quadrilateral,3,lobatto; hexahedron,1,lobatto; hexahedron,2,lobatto
Basix.UFL	`basix.ElementFamily.P` ↓ 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 Lagrange (equispaced variant) degree 1 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 3 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 3 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 3 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 3, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a hexahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.hexahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a hexahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.hexahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 1 on a prism element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.prism, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (equispaced variant) degree 2 on a prism element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.prism, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Lagrange (GLL variant) degree 1 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 2 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 3 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 3, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 4 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 4, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 1 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.gll_warped) # Create Lagrange (GLL variant) degree 2 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.gll_warped) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,1,equispaced; interval,2,equispaced; interval,3,equispaced; triangle,1,equispaced; triangle,2,equispaced; triangle,3,equispaced; quadrilateral,1,equispaced; quadrilateral,2,equispaced; quadrilateral,3,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced; prism,1,equispaced; prism,2,equispaced; interval,1,gll; interval,2,gll; interval,3,gll; interval,4,gll; quadrilateral,1,gll; quadrilateral,2,gll Not implemented: pyramid,1,equispaced; pyramid,2,equispaced; interval,1,lobatto; interval,2,lobatto; interval,3,lobatto; quadrilateral,1,lobatto; quadrilateral,2,lobatto; quadrilateral,3,lobatto; hexahedron,1,lobatto; hexahedron,2,lobatto
Bempp	`"P"` ↓ Show Bempp examples ↓↑ Hide Bempp examples ↑ Before running this example, you must install Bempp: pip3 install numba scipy meshio 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 degree 1 element = bempp.api.function_space(grid, "P", 1)
FIAT	`FIAT.Lagrange` ↓ 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 Lagrange (equispaced variant) degree 1 element = FIAT.Lagrange(FIAT.ufc_cell("interval"), 1) # Create Lagrange (equispaced variant) degree 2 element = FIAT.Lagrange(FIAT.ufc_cell("interval"), 2) # Create Lagrange (equispaced variant) degree 3 element = FIAT.Lagrange(FIAT.ufc_cell("interval"), 3) # Create Lagrange (equispaced variant) degree 1 element = FIAT.Lagrange(FIAT.ufc_cell("triangle"), 1) # Create Lagrange (equispaced variant) degree 2 element = FIAT.Lagrange(FIAT.ufc_cell("triangle"), 2) # Create Lagrange (equispaced variant) degree 3 element = FIAT.Lagrange(FIAT.ufc_cell("triangle"), 3) # Create Lagrange (equispaced variant) degree 1 element = FIAT.Lagrange(FIAT.ufc_cell("tetrahedron"), 1) # Create Lagrange (equispaced variant) degree 2 element = FIAT.Lagrange(FIAT.ufc_cell("tetrahedron"), 2) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,1,equispaced; interval,2,equispaced; interval,3,equispaced; triangle,1,equispaced; triangle,2,equispaced; triangle,3,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced Not implemented: quadrilateral,1,equispaced; quadrilateral,2,equispaced; quadrilateral,3,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced; prism,1,equispaced; prism,2,equispaced; pyramid,1,equispaced; pyramid,2,equispaced; interval,1,gll; interval,2,gll; interval,3,gll; interval,4,gll; quadrilateral,1,gll; quadrilateral,2,gll; interval,1,lobatto; interval,2,lobatto; interval,3,lobatto; quadrilateral,1,lobatto; quadrilateral,2,lobatto; quadrilateral,3,lobatto; hexahedron,1,lobatto; hexahedron,2,lobatto
NDElement	`Family.Lagrange` ↓ Show NDElement examples ↓↑ Hide NDElement examples ↑ Before running this example, you must install NDElement: pip3 install ndelement This element can then be created with the following lines of Python: from ndelement.ciarlet import Family, create_family from ndelement.reference_cell import ReferenceCellType # Create Lagrange (equispaced variant) degree 1 on a interval family = create_family(Family.Lagrange, 1) element = family.element(ReferenceCellType.Interval) # Create Lagrange (equispaced variant) degree 2 on a interval family = create_family(Family.Lagrange, 2) element = family.element(ReferenceCellType.Interval) # Create Lagrange (equispaced variant) degree 3 on a interval family = create_family(Family.Lagrange, 3) element = family.element(ReferenceCellType.Interval) # Create Lagrange (equispaced variant) degree 1 on a triangle family = create_family(Family.Lagrange, 1) element = family.element(ReferenceCellType.Triangle) # Create Lagrange (equispaced variant) degree 2 on a triangle family = create_family(Family.Lagrange, 2) element = family.element(ReferenceCellType.Triangle) # Create Lagrange (equispaced variant) degree 3 on a triangle family = create_family(Family.Lagrange, 3) element = family.element(ReferenceCellType.Triangle) # Create Lagrange (equispaced variant) degree 1 on a quadrilateral family = create_family(Family.Lagrange, 1) element = family.element(ReferenceCellType.Quadrilateral) # Create Lagrange (equispaced variant) degree 2 on a quadrilateral family = create_family(Family.Lagrange, 2) element = family.element(ReferenceCellType.Quadrilateral) # Create Lagrange (equispaced variant) degree 3 on a quadrilateral family = create_family(Family.Lagrange, 3) element = family.element(ReferenceCellType.Quadrilateral) # Create Lagrange (equispaced variant) degree 1 on a tetrahedron family = create_family(Family.Lagrange, 1) element = family.element(ReferenceCellType.Tetrahedron) # Create Lagrange (equispaced variant) degree 2 on a tetrahedron family = create_family(Family.Lagrange, 2) element = family.element(ReferenceCellType.Tetrahedron) # Create Lagrange (equispaced variant) degree 1 on a hexahedron family = create_family(Family.Lagrange, 1) element = family.element(ReferenceCellType.Hexahedron) # Create Lagrange (equispaced variant) degree 2 on a hexahedron family = create_family(Family.Lagrange, 2) element = family.element(ReferenceCellType.Hexahedron) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,1,equispaced; interval,2,equispaced; interval,3,equispaced; triangle,1,equispaced; triangle,2,equispaced; triangle,3,equispaced; quadrilateral,1,equispaced; quadrilateral,2,equispaced; quadrilateral,3,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced Not implemented: prism,1,equispaced; prism,2,equispaced; pyramid,1,equispaced; pyramid,2,equispaced; interval,1,gll; interval,2,gll; interval,3,gll; interval,4,gll; quadrilateral,1,gll; quadrilateral,2,gll; interval,1,lobatto; interval,2,lobatto; interval,3,lobatto; quadrilateral,1,lobatto; quadrilateral,2,lobatto; quadrilateral,3,lobatto; hexahedron,1,lobatto; hexahedron,2,lobatto
Symfem	`"Lagrange"` (interval, equispaced; triangle, equispaced; tetrahedron, equispaced; prism, equispaced; pyramid, equispaced) `"Q"` (quadrilateral, equispaced; hexahedron, equispaced) `"Lagrange", variant="gll"` (interval, GLL) `"Q", variant="gll"` (quadrilateral, GLL; hexahedron, GLL) `"Lagrange", variant="lobatto"` (interval, Lobatto) `"Q", variant="lobatto"` (quadrilateral, Lobatto; hexahedron, Lobatto) ↓ 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 Lagrange (equispaced variant) degree 1 on a interval element = symfem.create_element("interval", "Lagrange", 1) # Create Lagrange (equispaced variant) degree 2 on a interval element = symfem.create_element("interval", "Lagrange", 2) # Create Lagrange (equispaced variant) degree 3 on a interval element = symfem.create_element("interval", "Lagrange", 3) # Create Lagrange (equispaced variant) degree 1 on a triangle element = symfem.create_element("triangle", "Lagrange", 1) # Create Lagrange (equispaced variant) degree 2 on a triangle element = symfem.create_element("triangle", "Lagrange", 2) # Create Lagrange (equispaced variant) degree 3 on a triangle element = symfem.create_element("triangle", "Lagrange", 3) # Create Lagrange (equispaced variant) degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 1) # Create Lagrange (equispaced variant) degree 2 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 2) # Create Lagrange (equispaced variant) degree 3 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 3) # Create Lagrange (equispaced variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "Lagrange", 1) # Create Lagrange (equispaced variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "Lagrange", 2) # Create Lagrange (equispaced variant) degree 1 on a hexahedron element = symfem.create_element("hexahedron", "Q", 1) # Create Lagrange (equispaced variant) degree 2 on a hexahedron element = symfem.create_element("hexahedron", "Q", 2) # Create Lagrange (equispaced variant) degree 1 on a prism element = symfem.create_element("prism", "Lagrange", 1) # Create Lagrange (equispaced variant) degree 2 on a prism element = symfem.create_element("prism", "Lagrange", 2) # Create Lagrange (equispaced variant) degree 1 on a pyramid element = symfem.create_element("pyramid", "Lagrange", 1) # Create Lagrange (equispaced variant) degree 2 on a pyramid element = symfem.create_element("pyramid", "Lagrange", 2) # Create Lagrange (GLL variant) degree 1 on a interval element = symfem.create_element("interval", "Lagrange", 1, variant="gll") # Create Lagrange (GLL variant) degree 2 on a interval element = symfem.create_element("interval", "Lagrange", 2, variant="gll") # Create Lagrange (GLL variant) degree 3 on a interval element = symfem.create_element("interval", "Lagrange", 3, variant="gll") # Create Lagrange (GLL variant) degree 4 on a interval element = symfem.create_element("interval", "Lagrange", 4, variant="gll") # Create Lagrange (GLL variant) degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 1, variant="gll") # Create Lagrange (GLL variant) degree 2 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 2, variant="gll") # Create Lagrange (Lobatto variant) degree 1 on a interval element = symfem.create_element("interval", "Lagrange", 1, variant="lobatto") # Create Lagrange (Lobatto variant) degree 2 on a interval element = symfem.create_element("interval", "Lagrange", 2, variant="lobatto") # Create Lagrange (Lobatto variant) degree 3 on a interval element = symfem.create_element("interval", "Lagrange", 3, variant="lobatto") # Create Lagrange (Lobatto variant) degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 1, variant="lobatto") # Create Lagrange (Lobatto variant) degree 2 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 2, variant="lobatto") # Create Lagrange (Lobatto variant) degree 3 on a quadrilateral element = symfem.create_element("quadrilateral", "Q", 3, variant="lobatto") # Create Lagrange (Lobatto variant) degree 1 on a hexahedron element = symfem.create_element("hexahedron", "Q", 1, variant="lobatto") # Create Lagrange (Lobatto variant) degree 2 on a hexahedron element = symfem.create_element("hexahedron", "Q", 2, variant="lobatto") This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"Lagrange"` (interval, equispaced; triangle, equispaced; tetrahedron, equispaced) `"Q"` (quadrilateral, equispaced; hexahedron, equispaced) `"Lobatto"` (interval, Lobatto) ↓ 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 Lagrange (equispaced variant) degree 1 on a interval element = ufl_legacy.FiniteElement("Lagrange", "interval", 1) # Create Lagrange (equispaced variant) degree 2 on a interval element = ufl_legacy.FiniteElement("Lagrange", "interval", 2) # Create Lagrange (equispaced variant) degree 3 on a interval element = ufl_legacy.FiniteElement("Lagrange", "interval", 3) # Create Lagrange (equispaced variant) degree 1 on a triangle element = ufl_legacy.FiniteElement("Lagrange", "triangle", 1) # Create Lagrange (equispaced variant) degree 2 on a triangle element = ufl_legacy.FiniteElement("Lagrange", "triangle", 2) # Create Lagrange (equispaced variant) degree 3 on a triangle element = ufl_legacy.FiniteElement("Lagrange", "triangle", 3) # Create Lagrange (equispaced variant) degree 1 on a quadrilateral element = ufl_legacy.FiniteElement("Q", "quadrilateral", 1) # Create Lagrange (equispaced variant) degree 2 on a quadrilateral element = ufl_legacy.FiniteElement("Q", "quadrilateral", 2) # Create Lagrange (equispaced variant) degree 3 on a quadrilateral element = ufl_legacy.FiniteElement("Q", "quadrilateral", 3) # Create Lagrange (equispaced variant) degree 1 on a tetrahedron element = ufl_legacy.FiniteElement("Lagrange", "tetrahedron", 1) # Create Lagrange (equispaced variant) degree 2 on a tetrahedron element = ufl_legacy.FiniteElement("Lagrange", "tetrahedron", 2) # Create Lagrange (equispaced variant) degree 1 on a hexahedron element = ufl_legacy.FiniteElement("Q", "hexahedron", 1) # Create Lagrange (equispaced variant) degree 2 on a hexahedron element = ufl_legacy.FiniteElement("Q", "hexahedron", 2) # Create Lagrange (Lobatto variant) degree 1 on a interval element = ufl_legacy.FiniteElement("Lobatto", "interval", 1) # Create Lagrange (Lobatto variant) degree 2 on a interval element = ufl_legacy.FiniteElement("Lobatto", "interval", 2) # Create Lagrange (Lobatto variant) degree 3 on a interval element = ufl_legacy.FiniteElement("Lobatto", "interval", 3)

Examples

interval degree 1 equispaced variant	(click to view basis functions)
interval degree 2 equispaced variant	(click to view basis functions)
interval degree 3 equispaced variant	(click to view basis functions)
triangle degree 1 equispaced variant	(click to view basis functions)
triangle degree 2 equispaced variant	(click to view basis functions)
triangle degree 3 equispaced variant	(click to view basis functions)
quadrilateral degree 1 equispaced variant	(click to view basis functions)
quadrilateral degree 2 equispaced variant	(click to view basis functions)
quadrilateral degree 3 equispaced variant	(click to view basis functions)
tetrahedron degree 1 equispaced variant	(click to view basis functions)
tetrahedron degree 2 equispaced variant	(click to view basis functions)
hexahedron degree 1 equispaced variant	(click to view basis functions)
hexahedron degree 2 equispaced variant	(click to view basis functions)
prism degree 1 equispaced variant	(click to view basis functions)
prism degree 2 equispaced variant	(click to view basis functions)
pyramid degree 1 equispaced variant	(click to view basis functions)
pyramid degree 2 equispaced variant	(click to view basis functions)
interval degree 1 GLL variant	(click to view basis functions)
interval degree 2 GLL variant	(click to view basis functions)
interval degree 3 GLL variant	(click to view basis functions)
interval degree 4 GLL variant	(click to view basis functions)
quadrilateral degree 1 GLL variant	(click to view basis functions)
quadrilateral degree 2 GLL variant	(click to view basis functions)
interval degree 1 Lobatto variant	(click to view basis functions)
interval degree 2 Lobatto variant	(click to view basis functions)
interval degree 3 Lobatto variant	(click to view basis functions)
quadrilateral degree 1 Lobatto variant	(click to view basis functions)
quadrilateral degree 2 Lobatto variant	(click to view basis functions)
quadrilateral degree 3 Lobatto variant	(click to view basis functions)
hexahedron degree 1 Lobatto variant	(click to view basis functions)
hexahedron degree 2 Lobatto variant	(click to view basis functions)

References

Arnold, Douglas N. and Logg, Anders. Periodic table of the finite elements, SIAM News 47, 2014. [www.siam.org/publications/siam-news/issues/volume-47-number-09-november-2014/] [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	14 April 2021
Element last updated	16 October 2024