Discontinuous Lagrange

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Alternative names	discontinuous Galerkin, discontinuous Gauss–Lobatto–Legendre (GLL variant), Gauss–Legendre (GL variant), Legendre (Legendre variant)
De Rham complex families	\(\left[S_{2,k}^\unicode{0x25FA}\right]_{d}\) 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	DP, DQ (quadrilateral and hexahedron), DG, discontinuous GLL (GLL variant), GL (GL variant)
Variants	equispaced: The variant has its point evaluations at equally spaced points. GLL: This variant has its point evaluations at GLL points. GL: This variant has its point evaluations at GL points. Legendre: The basis functions of this variant are orthonormal Legendre polynomials.
Degrees	\(0\leqslant k\) 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) \(\mathcal{Z}^{(13)}_{k}\) (prism) \(\mathcal{P}_{k} \oplus \mathcal{Z}^{(14)}_{k}\) (pyramid) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)\leqslant k\right\}\) \(\mathcal{Z}^{(13)}_k=\operatorname{span}\left\{x_1^{p_1}x_2^{p_2}x_3^{p_3}\middle\|\max(p_1+p_2,p_3)\leqslant k\right\}\) \(\mathcal{Z}^{(14)}_k=\operatorname{span}\left\{\frac{x_1^{p_1}x_2^{p_2}}{(1-x_3)^{\min(p_1,p_2)}}\middle\|(p_1,p_2)=(1,k)\text{ or }(p_1,p_2)=(k,1)\text{ or }p_1\leqslant k-1,p_2\leqslant k-1,k+1\leqslant p_1+p_2\right\}\)
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	cells: \(k+1\) (A000027) (interval), \((k+1)(k+2)/2\) (A000217) (triangle), \((k+1)(k+2)(k+3)/6\) (A000292) (tetrahedron), \((k+1)^2\) (A000290) (quadrilateral), \((k+1)^3\) (A000578) (hexahedron), \((k+1)^2(k+2)/2\) (A002411) (prism), \((k+1)(k+2)(2k+3)/6\) (A000330) (pyramid)
Mapping	L2 Piola
continuity	Discontinuous.
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 discontinuous Lagrange (equispaced variant) degree 0 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a interval element = basix.create_element(basix.ElementFamily.P, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.P, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a quadrilateral element = basix.create_element(basix.ElementFamily.P, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a tetrahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.tetrahedron, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a hexahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.hexahedron, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a hexahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.hexahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a hexahedron element = basix.create_element(basix.ElementFamily.P, basix.CellType.hexahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a prism element = basix.create_element(basix.ElementFamily.P, basix.CellType.prism, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a prism element = basix.create_element(basix.ElementFamily.P, basix.CellType.prism, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a prism element = basix.create_element(basix.ElementFamily.P, basix.CellType.prism, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,0,equispaced; interval,1,equispaced; interval,2,equispaced; triangle,0,equispaced; triangle,1,equispaced; triangle,2,equispaced; quadrilateral,0,equispaced; quadrilateral,1,equispaced; quadrilateral,2,equispaced; tetrahedron,0,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced; hexahedron,0,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced; prism,0,equispaced; prism,1,equispaced; prism,2,equispaced Not implemented: pyramid,0,equispaced; pyramid,1,equispaced; pyramid,2,equispaced
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 discontinuous Lagrange (equispaced variant) degree 0 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a interval element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.tetrahedron, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a hexahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.hexahedron, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a hexahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.hexahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a hexahedron element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.hexahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 0 on a prism element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.prism, 0, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 1 on a prism element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.prism, 1, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) # Create discontinuous Lagrange (equispaced variant) degree 2 on a prism element = basix.ufl.element(basix.ElementFamily.P, basix.CellType.prism, 2, lagrange_variant=basix.LagrangeVariant.equispaced, discontinuous=True) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,0,equispaced; interval,1,equispaced; interval,2,equispaced; triangle,0,equispaced; triangle,1,equispaced; triangle,2,equispaced; quadrilateral,0,equispaced; quadrilateral,1,equispaced; quadrilateral,2,equispaced; tetrahedron,0,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced; hexahedron,0,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced; prism,0,equispaced; prism,1,equispaced; prism,2,equispaced Not implemented: pyramid,0,equispaced; pyramid,1,equispaced; pyramid,2,equispaced
Bempp	`"DP"` ↓ 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 discontinuous Lagrange degree 0 element = bempp.api.function_space(grid, "DP", 0) # Create discontinuous Lagrange degree 1 element = bempp.api.function_space(grid, "DP", 1)
FIAT	`FIAT.DiscontinuousLagrange` ↓ 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 discontinuous Lagrange (equispaced variant) degree 0 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("interval"), 0) # Create discontinuous Lagrange (equispaced variant) degree 1 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("interval"), 1) # Create discontinuous Lagrange (equispaced variant) degree 2 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("interval"), 2) # Create discontinuous Lagrange (equispaced variant) degree 0 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("triangle"), 0) # Create discontinuous Lagrange (equispaced variant) degree 1 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("triangle"), 1) # Create discontinuous Lagrange (equispaced variant) degree 2 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("triangle"), 2) # Create discontinuous Lagrange (equispaced variant) degree 0 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("tetrahedron"), 0) # Create discontinuous Lagrange (equispaced variant) degree 1 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("tetrahedron"), 1) # Create discontinuous Lagrange (equispaced variant) degree 2 element = FIAT.DiscontinuousLagrange(FIAT.ufc_cell("tetrahedron"), 2) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,0,equispaced; interval,1,equispaced; interval,2,equispaced; triangle,0,equispaced; triangle,1,equispaced; triangle,2,equispaced; tetrahedron,0,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced Not implemented: quadrilateral,0,equispaced; quadrilateral,1,equispaced; quadrilateral,2,equispaced; hexahedron,0,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced; prism,0,equispaced; prism,1,equispaced; prism,2,equispaced; pyramid,0,equispaced; pyramid,1,equispaced; pyramid,2,equispaced
NDElement	`Family.Lagrange, continuity=Continuity.Discontinuous` ↓ 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 Continuity, Family, create_family from ndelement.reference_cell import ReferenceCellType # Create discontinuous Lagrange (equispaced variant) degree 0 on a interval family = create_family(Family.Lagrange, 0, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Interval) # Create discontinuous Lagrange (equispaced variant) degree 1 on a interval family = create_family(Family.Lagrange, 1, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Interval) # Create discontinuous Lagrange (equispaced variant) degree 2 on a interval family = create_family(Family.Lagrange, 2, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Interval) # Create discontinuous Lagrange (equispaced variant) degree 0 on a triangle family = create_family(Family.Lagrange, 0, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Triangle) # Create discontinuous Lagrange (equispaced variant) degree 1 on a triangle family = create_family(Family.Lagrange, 1, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Triangle) # Create discontinuous Lagrange (equispaced variant) degree 2 on a triangle family = create_family(Family.Lagrange, 2, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Triangle) # Create discontinuous Lagrange (equispaced variant) degree 0 on a quadrilateral family = create_family(Family.Lagrange, 0, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Quadrilateral) # Create discontinuous Lagrange (equispaced variant) degree 1 on a quadrilateral family = create_family(Family.Lagrange, 1, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Quadrilateral) # Create discontinuous Lagrange (equispaced variant) degree 2 on a quadrilateral family = create_family(Family.Lagrange, 2, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Quadrilateral) # Create discontinuous Lagrange (equispaced variant) degree 0 on a tetrahedron family = create_family(Family.Lagrange, 0, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Tetrahedron) # Create discontinuous Lagrange (equispaced variant) degree 1 on a tetrahedron family = create_family(Family.Lagrange, 1, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Tetrahedron) # Create discontinuous Lagrange (equispaced variant) degree 2 on a tetrahedron family = create_family(Family.Lagrange, 2, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Tetrahedron) # Create discontinuous Lagrange (equispaced variant) degree 0 on a hexahedron family = create_family(Family.Lagrange, 0, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Hexahedron) # Create discontinuous Lagrange (equispaced variant) degree 1 on a hexahedron family = create_family(Family.Lagrange, 1, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Hexahedron) # Create discontinuous Lagrange (equispaced variant) degree 2 on a hexahedron family = create_family(Family.Lagrange, 2, continuity=Continuity.Discontinuous) element = family.element(ReferenceCellType.Hexahedron) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: interval,0,equispaced; interval,1,equispaced; interval,2,equispaced; triangle,0,equispaced; triangle,1,equispaced; triangle,2,equispaced; quadrilateral,0,equispaced; quadrilateral,1,equispaced; quadrilateral,2,equispaced; tetrahedron,0,equispaced; tetrahedron,1,equispaced; tetrahedron,2,equispaced; hexahedron,0,equispaced; hexahedron,1,equispaced; hexahedron,2,equispaced Not implemented: prism,0,equispaced; prism,1,equispaced; prism,2,equispaced; pyramid,0,equispaced; pyramid,1,equispaced; pyramid,2,equispaced
Symfem	`"discontinuous Lagrange"` (equispaced) `"discontinuous Lagrange", variant="gll"` (interval, GLL) `"discontinuous Q", variant="gll"` (quadrilateral, GLL; hexahedron, GLL) `"discontinuous Lagrange", variant="gl"` (interval, GL) `"discontinuous Q", variant="gl"` (quadrilateral, GL; hexahedron, GL) `"discontinuous Lagrange", variant="legendre"` (Legendre) ↓ 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 discontinuous Lagrange (equispaced variant) degree 0 on a interval element = symfem.create_element("interval", "discontinuous Lagrange", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a interval element = symfem.create_element("interval", "discontinuous Lagrange", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a interval element = symfem.create_element("interval", "discontinuous Lagrange", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a triangle element = symfem.create_element("triangle", "discontinuous Lagrange", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a triangle element = symfem.create_element("triangle", "discontinuous Lagrange", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a triangle element = symfem.create_element("triangle", "discontinuous Lagrange", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a quadrilateral element = symfem.create_element("quadrilateral", "discontinuous Lagrange", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "discontinuous Lagrange", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a quadrilateral element = symfem.create_element("quadrilateral", "discontinuous Lagrange", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a tetrahedron element = symfem.create_element("tetrahedron", "discontinuous Lagrange", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "discontinuous Lagrange", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "discontinuous Lagrange", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a hexahedron element = symfem.create_element("hexahedron", "discontinuous Lagrange", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a hexahedron element = symfem.create_element("hexahedron", "discontinuous Lagrange", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a hexahedron element = symfem.create_element("hexahedron", "discontinuous Lagrange", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a prism element = symfem.create_element("prism", "discontinuous Lagrange", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a prism element = symfem.create_element("prism", "discontinuous Lagrange", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a prism element = symfem.create_element("prism", "discontinuous Lagrange", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a pyramid element = symfem.create_element("pyramid", "discontinuous Lagrange", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a pyramid element = symfem.create_element("pyramid", "discontinuous Lagrange", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a pyramid element = symfem.create_element("pyramid", "discontinuous Lagrange", 2) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"Discontinuous Lagrange"` (interval, equispaced; triangle, equispaced; tetrahedron, equispaced) `"DQ"` (quadrilateral, equispaced; hexahedron, equispaced) ↓ 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 discontinuous Lagrange (equispaced variant) degree 0 on a interval element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "interval", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a interval element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "interval", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a interval element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "interval", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a triangle element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "triangle", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a triangle element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "triangle", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a triangle element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "triangle", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a quadrilateral element = ufl_legacy.FiniteElement("DQ", "quadrilateral", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a quadrilateral element = ufl_legacy.FiniteElement("DQ", "quadrilateral", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a quadrilateral element = ufl_legacy.FiniteElement("DQ", "quadrilateral", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a tetrahedron element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "tetrahedron", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a tetrahedron element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "tetrahedron", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a tetrahedron element = ufl_legacy.FiniteElement("Discontinuous Lagrange", "tetrahedron", 2) # Create discontinuous Lagrange (equispaced variant) degree 0 on a hexahedron element = ufl_legacy.FiniteElement("DQ", "hexahedron", 0) # Create discontinuous Lagrange (equispaced variant) degree 1 on a hexahedron element = ufl_legacy.FiniteElement("DQ", "hexahedron", 1) # Create discontinuous Lagrange (equispaced variant) degree 2 on a hexahedron element = ufl_legacy.FiniteElement("DQ", "hexahedron", 2)

Examples

interval degree 0 equispaced variant	(click to view basis functions)
interval degree 1 equispaced variant	(click to view basis functions)
interval degree 2 equispaced variant	(click to view basis functions)
triangle degree 0 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)
quadrilateral degree 0 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)
tetrahedron degree 0 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 0 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 0 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 0 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)

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]

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Element added	10 March 2025
Element last updated	10 March 2025