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Discontinuous Lagrange

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Alternative namesdiscontinuous 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 namesDP, DQ (quadrilateral and hexahedron), DG, discontinuous GLL (GLL variant), GL (GL variant)
Variantsequispaced: 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 superdegreeinterval: \(k\)
triangle: \(k\)
tetrahedron: \(k\)
quadrilateral: \(dk\)
hexahedron: \(dk\)
prism: \(2k\)
pyramid: undefined
Lagrange subdegree\(k\)
Lagrange superdegree\(k\)
Reference elementsinterval, 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 ↓
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 subentitiescells: \(k+1\) (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)
MappingL2 Piola
continuityDiscontinuous.
CategoriesScalar-valued elements

Implementations

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

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

DefElement stats

Element added10 March 2025
Element last updated24 March 2025