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Lagrange

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Alternative namesPolynomial, Galerkin, DGT (facets), Hdiv trace (facets), Q (quadrilateral and hexahedron), Gauss–Lobatto–Legendre (GLL variant), Lobatto (Lobatto variant)
De Rham complex families\(\left[S_{2,k}^\unicode{0x25FA}\right]_{0}\) / \(\mathcal{P}^-_{k}\Lambda^{0}(\Delta_d)\), \(\left[S_{1,k}^\unicode{0x25FA}\right]_{0}\) / \(\mathcal{P}_{k}\Lambda^{0}(\Delta_d)\), \(\left[S_{4,k}^\square\right]_{0}\) / \(\mathcal{Q}^-_{k}\Lambda^{0}(\square_d)\), \(\left[S_{2,k}^\unicode{0x25FA}\right]_{d}\) / \(\mathcal{P}^-_{k}\Lambda^{d}(\Delta_d)\), \(\left[S_{1,k}^\unicode{0x25FA}\right]_{d}\) / \(\mathcal{P}_{k}\Lambda^{d}(\Delta_d)\), \(\left[S_{4,k}^\square\right]_{d}\) / \(\mathcal{Q}^-_{k}\Lambda^{d}(\square_d)\), \(\left[S_{3,k}^\square\right]_{d}\)
Abbreviated namesP, CG, DG, GLL (GLL variant)
Variantsequispaced: 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
Orders\(1\leqslant k\)
Reference elementsinterval, triangle, tetrahedron, quadrilateral, hexahedron, prism, pyramid
Polynomial set\(\mathcal{P}_{k}\) (interval, triangle, tetrahedron)
\(\mathcal{Q}_{k}\) (quadrilateral, hexahedron)
\(\mathcal{P}_{k} \oplus \mathcal{Z}^{(15)}_{k}\) (prism)
\(\mathcal{Z}^{(16)}_{k} \oplus \mathcal{Z}^{(17)}_{k}\) (pyramid)
↓ Show polynomial set definitions ↓
DOFsOn each vertex: point evaluations
On each edge: point evaluations
On each face: point evaluations
On each volume: point evaluations
Number of DOFsinterval: \(k+1\) (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 subentitiesvertices: \(1\) (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)
Mappingidentity
continuityFunction values are continuous.
NotesDGT 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.
CategoriesScalar-valued elements

Implementations

Basixbasix.ElementFamily.P
↓ Show Basix examples ↓ This implementation is correct for all the examples below that it supports.
Basix.UFLbasix.ElementFamily.P
↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below that it supports.
Bempp"P"
↓ Show Bempp examples ↓
FIATFIAT.Lagrange
↓ Show FIAT examples ↓ This implementation is correct for all the examples below that it supports.
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 ↓ 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 ↓

Examples

interval
order 1
equispaced variant

(click to view basis functions)
interval
order 2
equispaced variant

(click to view basis functions)
interval
order 3
equispaced variant

(click to view basis functions)
triangle
order 1
equispaced variant

(click to view basis functions)
triangle
order 2
equispaced variant

(click to view basis functions)
triangle
order 3
equispaced variant

(click to view basis functions)
quadrilateral
order 1
equispaced variant

(click to view basis functions)
quadrilateral
order 2
equispaced variant

(click to view basis functions)
quadrilateral
order 3
equispaced variant

(click to view basis functions)
tetrahedron
order 1
equispaced variant

(click to view basis functions)
tetrahedron
order 2
equispaced variant

(click to view basis functions)
hexahedron
order 1
equispaced variant

(click to view basis functions)
hexahedron
order 2
equispaced variant

(click to view basis functions)
prism
order 1
equispaced variant

(click to view basis functions)
prism
order 2
equispaced variant

(click to view basis functions)
pyramid
order 1
equispaced variant

(click to view basis functions)
pyramid
order 2
equispaced variant

(click to view basis functions)
interval
order 1
GLL variant

(click to view basis functions)
interval
order 2
GLL variant

(click to view basis functions)
interval
order 3
GLL variant

(click to view basis functions)
interval
order 4
GLL variant

(click to view basis functions)
quadrilateral
order 1
GLL variant

(click to view basis functions)
quadrilateral
order 2
GLL variant

(click to view basis functions)
interval
order 1
Lobatto variant

(click to view basis functions)
interval
order 2
Lobatto variant

(click to view basis functions)
interval
order 3
Lobatto variant

(click to view basis functions)
quadrilateral
order 1
Lobatto variant

(click to view basis functions)
quadrilateral
order 2
Lobatto variant

(click to view basis functions)
quadrilateral
order 3
Lobatto variant

(click to view basis functions)
hexahedron
order 1
Lobatto variant

(click to view basis functions)
hexahedron
order 2
Lobatto variant

(click to view basis functions)

References

DefElement stats

Element added16 January 2021
Element last updated13 June 2024