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Lagrange

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Alternative namesPolynomial, Galerkin, DGT, Hdiv trace, Q (quadrilateral and hexahedron), \(\mathcal{P}^-_k\Lambda^{0}(\Delta_d)\), \(\mathcal{P}_k\Lambda^{0}(\Delta_d)\), \(\mathcal{Q}^-_k\Lambda^{0}(\square_d)\)
Abbreviated namesP, CG, DG
Orders\(1\leqslant k\)
Reference elementsinterval, triangle, tetrahedron, quadrilateral, hexahedron, prism, pyramid
Polynomial set\(\mathcal{P}_{k}\) (interval, triangle, tetrahedron)
\(\mathcal{Q}_{k}\) (quadrilateral, hexahedron)
\(\mathcal{Z}^{(13)}_{k}\) (prism)
\(\mathcal{Z}^{(14)}_{k} \oplus \mathcal{Z}^{(15)}_{k}\) (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)
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, prism, pyramid)
"Q" (quadrilateral, hexahedron)
↓ Show Symfem examples ↓
Basix string"Lagrange"
↓ Show Basix examples ↓
UFL string"Lagrange" (interval, triangle, tetrahedron)
"Q" (quadrilateral, hexahedron)
↓ Show UFL examples ↓
Bempp string"P" (triangle)
↓ Show Bempp 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
tetrahedron
order 1
tetrahedron
order 2
hexahedron
order 1
hexahedron
order 2
prism
order 1
prism
order 2
pyramid
order 1
pyramid
order 2
  • \(R\) 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.
  • \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(y\), \(z\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)

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

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

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

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

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

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

\(\displaystyle \phi_{3} = z\)

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

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

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

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

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

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

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

\(\displaystyle \phi_{3} = z \left(2 z - 1\right)\)

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

\(\displaystyle \phi_{4} = 4 y z\)

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

\(\displaystyle \phi_{5} = 4 x z\)

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

\(\displaystyle \phi_{6} = 4 x y\)

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

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

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(0,\tfrac{1}{2},0)\)

\(\displaystyle \phi_{8} = 4 y \left(- x - y - z + 1\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{1}{2},0,0)\)

\(\displaystyle \phi_{9} = 4 x \left(- x - y - z + 1\right)\)

This DOF is associated with edge 5 of the reference element.
  • \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(z\), \(y\), \(y z\), \(x\), \(x z\), \(x y\), \(x y z\)
  • \(\mathcal{L}=\{l_0,...,l_{7}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)

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

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

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

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

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

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

\(\displaystyle \phi_{3} = x y \left(1 - z\right)\)

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

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

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(1,0,1)\)

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

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1,1)\)

\(\displaystyle \phi_{6} = y z \left(1 - x\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(1,1,1)\)

\(\displaystyle \phi_{7} = x y z\)

This DOF is associated with vertex 7 of the reference element.
  • \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(z\), \(z^{2}\), \(y\), \(y z\), \(y z^{2}\), \(y^{2}\), \(y^{2} z\), \(y^{2} z^{2}\), \(x\), \(x z\), \(x z^{2}\), \(x y\), \(x y z\), \(x y z^{2}\), \(x y^{2}\), \(x y^{2} z\), \(x y^{2} z^{2}\), \(x^{2}\), \(x^{2} z\), \(x^{2} z^{2}\), \(x^{2} y\), \(x^{2} y z\), \(x^{2} y z^{2}\), \(x^{2} y^{2}\), \(x^{2} y^{2} z\), \(x^{2} y^{2} z^{2}\)
  • \(\mathcal{L}=\{l_0,...,l_{26}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)

\(\displaystyle \phi_{0} = 8 x^{2} y^{2} z^{2} - 12 x^{2} y^{2} z + 4 x^{2} y^{2} - 12 x^{2} y z^{2} + 18 x^{2} y z - 6 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} - 12 x y^{2} z^{2} + 18 x y^{2} z - 6 x y^{2} + 18 x y z^{2} - 27 x y z + 9 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\)

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

\(\displaystyle \phi_{1} = x \left(8 x y^{2} z^{2} - 12 x y^{2} z + 4 x y^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 4 x z^{2} - 6 x z + 2 x - 4 y^{2} z^{2} + 6 y^{2} z - 2 y^{2} + 6 y z^{2} - 9 y z + 3 y - 2 z^{2} + 3 z - 1\right)\)

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

\(\displaystyle \phi_{2} = y \left(8 x^{2} y z^{2} - 12 x^{2} y z + 4 x^{2} y - 4 x^{2} z^{2} + 6 x^{2} z - 2 x^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 6 x z^{2} - 9 x z + 3 x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\)

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

\(\displaystyle \phi_{3} = x y \left(8 x y z^{2} - 12 x y z + 4 x y - 4 x z^{2} + 6 x z - 2 x - 4 y z^{2} + 6 y z - 2 y + 2 z^{2} - 3 z + 1\right)\)

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

\(\displaystyle \phi_{4} = z \left(8 x^{2} y^{2} z - 4 x^{2} y^{2} - 12 x^{2} y z + 6 x^{2} y + 4 x^{2} z - 2 x^{2} - 12 x y^{2} z + 6 x y^{2} + 18 x y z - 9 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\)

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(1,0,1)\)

\(\displaystyle \phi_{5} = x z \left(8 x y^{2} z - 4 x y^{2} - 12 x y z + 6 x y + 4 x z - 2 x - 4 y^{2} z + 2 y^{2} + 6 y z - 3 y - 2 z + 1\right)\)

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1,1)\)

\(\displaystyle \phi_{6} = y z \left(8 x^{2} y z - 4 x^{2} y - 4 x^{2} z + 2 x^{2} - 12 x y z + 6 x y + 6 x z - 3 x + 4 y z - 2 y - 2 z + 1\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(1,1,1)\)

\(\displaystyle \phi_{7} = x y z \left(8 x y z - 4 x y - 4 x z + 2 x - 4 y z + 2 y + 2 z - 1\right)\)

This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{1}{2},0,0)\)

\(\displaystyle \phi_{8} = 4 x \left(- 4 x y^{2} z^{2} + 6 x y^{2} z - 2 x y^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 2 x z^{2} + 3 x z - x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(0,\tfrac{1}{2},0)\)

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

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(0,0,\tfrac{1}{2})\)

\(\displaystyle \phi_{10} = 4 z \left(- 4 x^{2} y^{2} z + 4 x^{2} y^{2} + 6 x^{2} y z - 6 x^{2} y - 2 x^{2} z + 2 x^{2} + 6 x y^{2} z - 6 x y^{2} - 9 x y z + 9 x y + 3 x z - 3 x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(1,\tfrac{1}{2},0)\)

\(\displaystyle \phi_{11} = 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 4 x z^{2} - 6 x z + 2 x + 2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(1,0,\tfrac{1}{2})\)

\(\displaystyle \phi_{12} = 4 x z \left(- 4 x y^{2} z + 4 x y^{2} + 6 x y z - 6 x y - 2 x z + 2 x + 2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{13}:v\mapsto v(\tfrac{1}{2},1,0)\)

\(\displaystyle \phi_{13} = 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 2 x z^{2} - 3 x z + x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{14}:v\mapsto v(0,1,\tfrac{1}{2})\)

\(\displaystyle \phi_{14} = 4 y z \left(- 4 x^{2} y z + 4 x^{2} y + 2 x^{2} z - 2 x^{2} + 6 x y z - 6 x y - 3 x z + 3 x - 2 y z + 2 y + z - 1\right)\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{15}:v\mapsto v(1,1,\tfrac{1}{2})\)

\(\displaystyle \phi_{15} = 4 x y z \left(- 4 x y z + 4 x y + 2 x z - 2 x + 2 y z - 2 y - z + 1\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{16}:v\mapsto v(\tfrac{1}{2},0,1)\)

\(\displaystyle \phi_{16} = 4 x z \left(- 4 x y^{2} z + 2 x y^{2} + 6 x y z - 3 x y - 2 x z + x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\)

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{17}:v\mapsto v(0,\tfrac{1}{2},1)\)

\(\displaystyle \phi_{17} = 4 y z \left(- 4 x^{2} y z + 2 x^{2} y + 4 x^{2} z - 2 x^{2} + 6 x y z - 3 x y - 6 x z + 3 x - 2 y z + y + 2 z - 1\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{18}:v\mapsto v(1,\tfrac{1}{2},1)\)

\(\displaystyle \phi_{18} = 4 x y z \left(- 4 x y z + 2 x y + 4 x z - 2 x + 2 y z - y - 2 z + 1\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{19}:v\mapsto v(\tfrac{1}{2},1,1)\)

\(\displaystyle \phi_{19} = 4 x y z \left(- 4 x y z + 2 x y + 2 x z - x + 4 y z - 2 y - 2 z + 1\right)\)

This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{20}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},0)\)

\(\displaystyle \phi_{20} = 16 x y \left(2 x y z^{2} - 3 x y z + x y - 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{21}:v\mapsto v(\tfrac{1}{2},0,\tfrac{1}{2})\)

\(\displaystyle \phi_{21} = 16 x z \left(2 x y^{2} z - 2 x y^{2} - 3 x y z + 3 x y + x z - x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{22}:v\mapsto v(0,\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{22} = 16 y z \left(2 x^{2} y z - 2 x^{2} y - 2 x^{2} z + 2 x^{2} - 3 x y z + 3 x y + 3 x z - 3 x + y z - y - z + 1\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{23}:v\mapsto v(1,\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{23} = 16 x y z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{24}:v\mapsto v(\tfrac{1}{2},1,\tfrac{1}{2})\)

\(\displaystyle \phi_{24} = 16 x y z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{25}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},1)\)

\(\displaystyle \phi_{25} = 16 x y z \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{26}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{26} = 64 x y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)

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

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

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

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

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

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

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

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

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

\(\displaystyle \phi_{4} = x z\)

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,1,1)\)

\(\displaystyle \phi_{5} = y z\)

This DOF is associated with vertex 5 of the reference element.
  • \(R\) is the reference prism. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\), \(z\), \(x z\), \(x^{2} z\), \(y z\), \(x y z\), \(y^{2} z\), \(z^{2}\), \(x z^{2}\), \(x^{2} z^{2}\), \(y z^{2}\), \(x y z^{2}\), \(y^{2} z^{2}\)
  • \(\mathcal{L}=\{l_0,...,l_{17}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)

\(\displaystyle \phi_{0} = 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} + 8 x y z^{2} - 12 x y z + 4 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\)

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

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

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

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

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

\(\displaystyle \phi_{3} = z \left(4 x^{2} z - 2 x^{2} + 8 x y z - 4 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\)

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

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

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,1,1)\)

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

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(\tfrac{1}{2},0,0)\)

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

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

\(\displaystyle \phi_{7} = 4 y \left(- 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(0,0,\tfrac{1}{2})\)

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

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},0)\)

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

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(1,0,\tfrac{1}{2})\)

\(\displaystyle \phi_{10} = 4 x z \left(- 2 x z + 2 x + z - 1\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(0,1,\tfrac{1}{2})\)

\(\displaystyle \phi_{11} = 4 y z \left(- 2 y z + 2 y + z - 1\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(\tfrac{1}{2},0,1)\)

\(\displaystyle \phi_{12} = 4 x z \left(- 2 x z + x - 2 y z + y + 2 z - 1\right)\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{13}:v\mapsto v(0,\tfrac{1}{2},1)\)

\(\displaystyle \phi_{13} = 4 y z \left(- 2 x z + x - 2 y z + y + 2 z - 1\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{14}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},1)\)

\(\displaystyle \phi_{14} = 4 x y z \left(2 z - 1\right)\)

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{15}:v\mapsto v(\tfrac{1}{2},0,\tfrac{1}{2})\)

\(\displaystyle \phi_{15} = 16 x z \left(x z - x + y z - y - z + 1\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{16}:v\mapsto v(0,\tfrac{1}{2},\tfrac{1}{2})\)

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

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{17}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{17} = 16 x y z \left(1 - z\right)\)

This DOF is associated with face 3 of the reference element.
  • \(R\) is the reference pyramid. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1 - z\), \(y\), \(x\), \(- \frac{x y}{z - 1}\), \(z\)
  • \(\mathcal{L}=\{l_0,...,l_{4}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)

\(\displaystyle \phi_{0} = \frac{- x y + \left(z - 1\right) \left(- x - y - z + 1\right)}{z - 1}\)

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

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

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

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

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

\(\displaystyle \phi_{3} = - \frac{x y}{z - 1}\)

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

\(\displaystyle \phi_{4} = z\)

This DOF is associated with vertex 4 of the reference element.
  • \(R\) is the reference pyramid. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(z^{2} - 2 z + 1\), \(y \left(1 - z\right)\), \(y^{2}\), \(x \left(1 - z\right)\), \(x y\), \(- \frac{x y^{2}}{z - 1}\), \(x^{2}\), \(- \frac{x^{2} y}{z - 1}\), \(\frac{x^{2} y^{2}}{z^{2} - 2 z + 1}\), \(z \left(1 - z\right)\), \(y z\), \(x z\), \(- \frac{x y z}{z - 1}\), \(z^{2}\)
  • \(\mathcal{L}=\{l_0,...,l_{13}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)

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

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

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

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

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

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

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

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

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

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(\tfrac{1}{2},0,0)\)

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

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

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

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

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

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(1,\tfrac{1}{2},0)\)

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

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{1}{2},0,\tfrac{1}{2})\)

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

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(\tfrac{1}{2},1,0)\)

\(\displaystyle \phi_{10} = \frac{4 x y \left(- 2 x y - x z + x - 2 y z + 2 y - z^{2} + 2 z - 1\right)}{z^{2} - 2 z + 1}\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(0,\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{11} = \frac{4 y z \left(x + z - 1\right)}{z - 1}\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{12} = - \frac{4 x y z}{z - 1}\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{13}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},0)\)

\(\displaystyle \phi_{13} = \frac{16 x y \left(x y + x z - x + y z - y + z^{2} - 2 z + 1\right)}{z^{2} - 2 z + 1}\)

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