an encyclopedia of finite element definitions

Lagrange

Click here to read what the information on this page means.

Alternative namesPolynomial, 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 namesP, CG, DG
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)
\(\mathcal{Z}^{(11)}_{k}\) (prism)
\(\mathcal{Z}^{(12)}_{k} \oplus \mathcal{Z}^{(13)}_{k}\) (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.
CategoriesScalar-valued elements

Implementations

Basixbasix.ElementFamily.P, ..., basix.LagrangeVariant.equispaced
↓ Show Basix examples ↓
Bempp"P" (triangle)
↓ Show Bempp examples ↓
Symfem"Lagrange" (interval, triangle, tetrahedron, prism, pyramid)
"Q" (quadrilateral, hexahedron)
↓ Show Symfem examples ↓
UFL"Lagrange" (interval, triangle, tetrahedron)
"Q" (quadrilateral, hexahedron)
↓ Show UFL examples ↓

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

DefElement stats

Element added30 December 2020
Element last updated02 August 2022