an encyclopedia of finite element definitions

Brezzi–Douglas–Fortin–Marini

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

Abbreviated namesBDFM
Orders\(1\leqslant k\)
Reference elementstriangle, quadrilateral, tetrahedron, hexahedron
Polynomial set\(\mathcal{Z}^{(38)}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with an order \(k-1\) Lagrange space
On the interior of the reference element: integral moments with an order \(k-2\) vector Lagrange space
Number of DOFstriangle: \(k(k+2)\)
quadrilateral: \(k(k+3)\) (A028552)
tetrahedron: \(k(k+1)(k+3)/2\)
hexahedron: \(k(k+1)(k+5)/2\)
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"BDFM"
↓ Show Symfem examples ↓
UFL"BDFM" (triangle, tetrahedron)
↓ Show UFL examples ↓

Examples

triangle
order 1
triangle
order 2
quadrilateral
order 1
quadrilateral
order 2
tetrahedron
order 2
hexahedron
order 2
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{2}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle - y\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle x - 1\\\displaystyle y\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle 1 - y\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{7}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 4 x \left(1 - 2 x\right)\\\displaystyle 2 y \left(1 - 4 x\right)\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 2 x \left(1 - 4 y\right)\\\displaystyle 4 y \left(1 - 2 y\right)\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 8 x^{2} - 8 x y + 12 x + 6 y - 4\\\displaystyle 2 y \left(- 4 x - 4 y + 3\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 8 x y - 2 x - 6 y + 2\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 x \left(4 x + 4 y - 3\right)\\\displaystyle 8 x y - 6 x + 8 y^{2} - 12 y + 4\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 4 x \left(1 - 2 x\right)\\\displaystyle - 8 x y + 6 x + 2 y - 2\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 8 x \left(- 2 x - y + 2\right)\\\displaystyle 8 y \left(- 2 x - y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 8 x \left(- x - 2 y + 1\right)\\\displaystyle 8 y \left(- x - 2 y + 2\right)\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 1 - y\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle x - 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\)

This DOF is associated with edge 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: \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{9}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x y - 6 x + 3 y^{2} - 7 y + 4\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x + 3 y^{2} - y - 2\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 3 x^{2} - 6 x y + 7 x + 6 y - 4\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 3 x^{2} + 6 x y + x - 6 y + 2\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle x \left(- 3 x + 6 y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle x \left(- 3 x - 6 y + 5\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(- 6 x + 3 y + 1\right)\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x + 3 y - 5\right)\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - x\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(1 - y\right)\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle y z\\\displaystyle z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\\\displaystyle x z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{14}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 6 y \left(5 x - 1\right)\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x \left(5 y - 1\right)\\\displaystyle 6 y \left(5 y - 2\right)\\\displaystyle 6 z \left(5 y - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 6 y \left(5 z - 1\right)\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 30 x y + 30 x z - 48 x - 24 y - 24 z + 18\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 z \left(5 x + 5 y + 5 z - 4\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 30 x y + 6 x + 24 y - 6\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle 6 z \left(1 - 5 y\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 30 x z + 6 x + 24 z - 6\\\displaystyle 6 y \left(1 - 5 z\right)\\\displaystyle 6 z \left(2 - 5 z\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 6 x \left(- 5 x - 5 y - 5 z + 4\right)\\\displaystyle - 30 x y + 24 x - 30 y^{2} - 30 y z + 48 y + 24 z - 18\\\displaystyle 6 z \left(- 5 x - 5 y - 5 z + 4\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 30 x y - 24 x - 6 y + 6\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 30 y z - 6 y - 24 z + 6\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 30 x z - 24 x + 30 y z - 24 y + 30 z^{2} - 48 z + 18\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 6 x \left(2 - 5 x\right)\\\displaystyle 6 y \left(1 - 5 x\right)\\\displaystyle - 30 x z + 24 x + 6 z - 6\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - 5 y\right)\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle - 30 y z + 24 y + 6 z - 6\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 30 x \left(- 2 x - y - z + 2\right)\\\displaystyle 30 y \left(- 2 x - y - z + 1\right)\\\displaystyle 30 z \left(- 2 x - y - z + 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - 2 y - z + 1\right)\\\displaystyle 30 y \left(- x - 2 y - z + 2\right)\\\displaystyle 30 z \left(- x - 2 y - z + 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - y - 2 z + 1\right)\\\displaystyle 30 y \left(- x - y - 2 z + 1\right)\\\displaystyle 30 z \left(- x - y - 2 z + 2\right)\end{array}\right)\)

This DOF is associated with volume 0 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{20}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x + 6 y z - 6 y + 3 z^{2} - 10 z + 7\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 6 x z + 6 x + 6 y z - 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x - 6 y z + 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} - 6 y z + 10 y + 6 z - 7\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x y - 6 x - 3 y^{2} - 6 y z + 4 y + 6 z - 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} + 6 y z + 4 y - 6 z - 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y + 6 x z - 10 x - 6 y - 6 z + 7\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 6 x y + 6 x z - 4 x + 6 y - 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y - 6 x z - 4 x - 6 y + 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle x \left(3 x + 6 y - 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(- 6 x - 3 y + 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y - 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x - 6 y + 3 z + 4\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(6 x - 6 y + 3 z - 2\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x + 6 y + 3 z - 2\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - x\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(1 - y\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 z \left(1 - z\right)\end{array}\right)\)

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

References

DefElement stats

Element added30 January 2021
Element last updated05 June 2021