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Lagrange

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Alternative namesPolynomial, Galerkin, DGT, Hdiv trace, Q (quadrilaterals and hexahedra), \(\mathcal{P}^-_k\Lambda^{0}(\Delta_d)\), \(\mathcal{P}_k\Lambda^{0}(\Delta_d)\), \(\mathcal{Q}^-_k\Lambda^{0}(\square_d)\)
Abbreviated namesP, CG, DG
Orders\(1\leqslant k\)
Reference elementsinterval, triangle, tetrahedron, quadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k}\) (interval, triangle, tetrahedron)
\(\mathcal{Q}_{k}\) (quadrilateral, hexahedron)
↓ 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)
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)
NotesDGT and Hdiv trace are names given to this element when it is defined on the facets of a mesh.
CategoriesScalar-valued elements

Implementations

Symfem string"Lagrange" (interval, triangle, tetrahedron)
"Q" (quadrilateral, hexahedron)
↓ Show Symfem examples ↓
Basix string"Lagrange"
↓ Show Basix examples ↓

Examples

interval
order 1
interval
order 2
interval
order 3
triangle
order 1
triangle
order 2
triangle
order 3
quadrilateral
order 1
quadrilateral
order 2
quadrilateral
order 3
  • \(R\) is the reference interval. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\)
  • \(\mathcal{L}=\{l_0,...,l_{1}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = 1 - x\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1)\)

\(\displaystyle \phi_{1} = x\)

This DOF is associated with vertex 1 of the reference element.
  • \(R\) is the reference interval. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\)
  • \(\mathcal{L}=\{l_0,...,l_{2}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = 2 x^{2} - 3 x + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1)\)

\(\displaystyle \phi_{1} = x \left(2 x - 1\right)\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{2})\)

\(\displaystyle \phi_{2} = 4 x \left(1 - x\right)\)

This DOF is associated with edge 0 of the reference element.
  • \(R\) is the reference interval. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(x^{3}\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} + 9 x^{2} - \frac{11 x}{2} + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1)\)

\(\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{3})\)

\(\displaystyle \phi_{2} = \frac{9 x \left(3 x^{2} - 5 x + 2\right)}{2}\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{2}{3})\)

\(\displaystyle \phi_{3} = \frac{9 x \left(- 3 x^{2} + 4 x - 1\right)}{2}\)

This DOF is associated with edge 0 of the reference element.
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(y\)
  • \(\mathcal{L}=\{l_0,...,l_{2}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = - x - y + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)

\(\displaystyle \phi_{1} = x\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)

\(\displaystyle \phi_{2} = y\)

This DOF is associated with vertex 2 of the reference element.
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\)
  • \(\mathcal{L}=\{l_0,...,l_{5}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = 2 x^{2} + 4 x y - 3 x + 2 y^{2} - 3 y + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)

\(\displaystyle \phi_{1} = x \left(2 x - 1\right)\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)

\(\displaystyle \phi_{2} = y \left(2 y - 1\right)\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{3} = 4 x y\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(0,\tfrac{1}{2})\)

\(\displaystyle \phi_{4} = 4 y \left(- x - y + 1\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(\tfrac{1}{2},0)\)

\(\displaystyle \phi_{5} = 4 x \left(- x - y + 1\right)\)

This DOF is associated with edge 2 of the reference element.
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(y\), \(x y\), \(x^{2} y\), \(y^{2}\), \(x y^{2}\), \(y^{3}\)
  • \(\mathcal{L}=\{l_0,...,l_{9}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)

\(\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)

\(\displaystyle \phi_{2} = \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{2}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{3} = \frac{9 x y \left(3 x - 1\right)}{2}\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})\)

\(\displaystyle \phi_{4} = \frac{9 x y \left(3 y - 1\right)}{2}\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{3})\)

\(\displaystyle \phi_{5} = \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,\tfrac{2}{3})\)

\(\displaystyle \phi_{6} = \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(\tfrac{1}{3},0)\)

\(\displaystyle \phi_{7} = \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{2}{3},0)\)

\(\displaystyle \phi_{8} = \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(y\), \(x\), \(x y\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = x y - x - y + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)

\(\displaystyle \phi_{1} = x \left(1 - y\right)\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)

\(\displaystyle \phi_{2} = y \left(1 - x\right)\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1)\)

\(\displaystyle \phi_{3} = x y\)

This DOF is associated with vertex 3 of the reference element.
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(y\), \(y^{2}\), \(x\), \(x y\), \(x y^{2}\), \(x^{2}\), \(x^{2} y\), \(x^{2} y^{2}\)
  • \(\mathcal{L}=\{l_0,...,l_{8}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = 4 x^{2} y^{2} - 6 x^{2} y + 2 x^{2} - 6 x y^{2} + 9 x y - 3 x + 2 y^{2} - 3 y + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)

\(\displaystyle \phi_{1} = x \left(4 x y^{2} - 6 x y + 2 x - 2 y^{2} + 3 y - 1\right)\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)

\(\displaystyle \phi_{2} = y \left(4 x^{2} y - 2 x^{2} - 6 x y + 3 x + 2 y - 1\right)\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1)\)

\(\displaystyle \phi_{3} = x y \left(4 x y - 2 x - 2 y + 1\right)\)

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{2},0)\)

\(\displaystyle \phi_{4} = 4 x \left(- 2 x y^{2} + 3 x y - x + 2 y^{2} - 3 y + 1\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{2})\)

\(\displaystyle \phi_{5} = 4 y \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(1,\tfrac{1}{2})\)

\(\displaystyle \phi_{6} = 4 x y \left(- 2 x y + 2 x + y - 1\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(\tfrac{1}{2},1)\)

\(\displaystyle \phi_{7} = 4 x y \left(- 2 x y + x + 2 y - 1\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{8} = 16 x y \left(x y - x - y + 1\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(y\), \(y^{2}\), \(y^{3}\), \(x\), \(x y\), \(x y^{2}\), \(x y^{3}\), \(x^{2}\), \(x^{2} y\), \(x^{2} y^{2}\), \(x^{2} y^{3}\), \(x^{3}\), \(x^{3} y\), \(x^{3} y^{2}\), \(x^{3} y^{3}\)
  • \(\mathcal{L}=\{l_0,...,l_{15}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = \frac{81 x^{3} y^{3}}{4} - \frac{81 x^{3} y^{2}}{2} + \frac{99 x^{3} y}{4} - \frac{9 x^{3}}{2} - \frac{81 x^{2} y^{3}}{2} + 81 x^{2} y^{2} - \frac{99 x^{2} y}{2} + 9 x^{2} + \frac{99 x y^{3}}{4} - \frac{99 x y^{2}}{2} + \frac{121 x y}{4} - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)

\(\displaystyle \phi_{1} = \frac{x \left(- 81 x^{2} y^{3} + 162 x^{2} y^{2} - 99 x^{2} y + 18 x^{2} + 81 x y^{3} - 162 x y^{2} + 99 x y - 18 x - 18 y^{3} + 36 y^{2} - 22 y + 4\right)}{4}\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)

\(\displaystyle \phi_{2} = \frac{y \left(- 81 x^{3} y^{2} + 81 x^{3} y - 18 x^{3} + 162 x^{2} y^{2} - 162 x^{2} y + 36 x^{2} - 99 x y^{2} + 99 x y - 22 x + 18 y^{2} - 18 y + 4\right)}{4}\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1)\)

\(\displaystyle \phi_{3} = \frac{x y \left(81 x^{2} y^{2} - 81 x^{2} y + 18 x^{2} - 81 x y^{2} + 81 x y - 18 x + 18 y^{2} - 18 y + 4\right)}{4}\)

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{3},0)\)

\(\displaystyle \phi_{4} = \frac{9 x \left(- 27 x^{2} y^{3} + 54 x^{2} y^{2} - 33 x^{2} y + 6 x^{2} + 45 x y^{3} - 90 x y^{2} + 55 x y - 10 x - 18 y^{3} + 36 y^{2} - 22 y + 4\right)}{4}\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(\tfrac{2}{3},0)\)

\(\displaystyle \phi_{5} = \frac{9 x \left(27 x^{2} y^{3} - 54 x^{2} y^{2} + 33 x^{2} y - 6 x^{2} - 36 x y^{3} + 72 x y^{2} - 44 x y + 8 x + 9 y^{3} - 18 y^{2} + 11 y - 2\right)}{4}\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,\tfrac{1}{3})\)

\(\displaystyle \phi_{6} = \frac{9 y \left(- 27 x^{3} y^{2} + 45 x^{3} y - 18 x^{3} + 54 x^{2} y^{2} - 90 x^{2} y + 36 x^{2} - 33 x y^{2} + 55 x y - 22 x + 6 y^{2} - 10 y + 4\right)}{4}\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(0,\tfrac{2}{3})\)

\(\displaystyle \phi_{7} = \frac{9 y \left(27 x^{3} y^{2} - 36 x^{3} y + 9 x^{3} - 54 x^{2} y^{2} + 72 x^{2} y - 18 x^{2} + 33 x y^{2} - 44 x y + 11 x - 6 y^{2} + 8 y - 2\right)}{4}\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(1,\tfrac{1}{3})\)

\(\displaystyle \phi_{8} = \frac{9 x y \left(27 x^{2} y^{2} - 45 x^{2} y + 18 x^{2} - 27 x y^{2} + 45 x y - 18 x + 6 y^{2} - 10 y + 4\right)}{4}\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(1,\tfrac{2}{3})\)

\(\displaystyle \phi_{9} = \frac{9 x y \left(- 27 x^{2} y^{2} + 36 x^{2} y - 9 x^{2} + 27 x y^{2} - 36 x y + 9 x - 6 y^{2} + 8 y - 2\right)}{4}\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(\tfrac{1}{3},1)\)

\(\displaystyle \phi_{10} = \frac{9 x y \left(27 x^{2} y^{2} - 27 x^{2} y + 6 x^{2} - 45 x y^{2} + 45 x y - 10 x + 18 y^{2} - 18 y + 4\right)}{4}\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(\tfrac{2}{3},1)\)

\(\displaystyle \phi_{11} = \frac{9 x y \left(- 27 x^{2} y^{2} + 27 x^{2} y - 6 x^{2} + 36 x y^{2} - 36 x y + 8 x - 9 y^{2} + 9 y - 2\right)}{4}\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{12} = \frac{81 x y \left(9 x^{2} y^{2} - 15 x^{2} y + 6 x^{2} - 15 x y^{2} + 25 x y - 10 x + 6 y^{2} - 10 y + 4\right)}{4}\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:v\mapsto v(\tfrac{2}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{13} = \frac{81 x y \left(- 9 x^{2} y^{2} + 15 x^{2} y - 6 x^{2} + 12 x y^{2} - 20 x y + 8 x - 3 y^{2} + 5 y - 2\right)}{4}\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})\)

\(\displaystyle \phi_{14} = \frac{81 x y \left(- 9 x^{2} y^{2} + 12 x^{2} y - 3 x^{2} + 15 x y^{2} - 20 x y + 5 x - 6 y^{2} + 8 y - 2\right)}{4}\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{15}:v\mapsto v(\tfrac{2}{3},\tfrac{2}{3})\)

\(\displaystyle \phi_{15} = \frac{81 x y \left(9 x^{2} y^{2} - 12 x^{2} y + 3 x^{2} - 12 x y^{2} + 16 x y - 4 x + 3 y^{2} - 4 y + 1\right)}{4}\)

This DOF is associated with face 0 of the reference element.