Degree 1 Tiniest tensor H(div) on a hexahedron

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In this example:

$R$ is the reference hexahedron. The following numbering of the subentities of the reference is used:

$V$ is spanned by: $(\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 1 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})$ , $(\begin{matrix} z \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ z \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ z \end{matrix})$ , $(\begin{matrix} y \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ y \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ y \end{matrix})$ , $(\begin{matrix} y z \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ y z \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ y z \end{matrix})$ , $(\begin{matrix} x \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ x \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ x \end{matrix})$ , $(\begin{matrix} x z \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ x z \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ x z \end{matrix})$ , $(\begin{matrix} x y \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ x y \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ x y \end{matrix})$ , $(\begin{matrix} x y z \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ x y z \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ x y z \end{matrix})$ , $(\begin{matrix} 0 \\ 0 \\ \frac{3 z (z - 1)}{2} \end{matrix})$ , $(\begin{matrix} 0 \\ \frac{3 y (y - 1)}{2} \\ 0 \end{matrix})$ , $(\begin{matrix} 0 \\ \frac{3 y (- 2 y z + y + 2 z - 1)}{2} \\ \frac{3 z (- 2 y z + 2 y + z - 1)}{2} \end{matrix})$ , $(\begin{matrix} \frac{3 x (x - 1)}{2} \\ 0 \\ 0 \end{matrix})$ , $(\begin{matrix} \frac{3 x (- 2 x z + x + 2 z - 1)}{2} \\ 0 \\ \frac{3 z (- 2 x z + 2 x + z - 1)}{2} \end{matrix})$ , $(\begin{matrix} \frac{3 x (- 2 x y + x + 2 y - 1)}{2} \\ \frac{3 y (- 2 x y + 2 x + y - 1)}{2} \\ 0 \end{matrix})$ , $(\begin{matrix} \frac{3 x (4 x y z - 2 x y - 2 x z + x - 4 y z + 2 y + 2 z - 1)}{2} \\ \frac{3 y (4 x y z - 2 x y - 4 x z + 2 x - 2 y z + y + 2 z - 1)}{2} \\ \frac{3 z (4 x y z - 4 x y - 2 x z + 2 x - 2 y z + 2 y + z - 1)}{2} \end{matrix})$
$L = {l_{0}, . . ., l_{30}}$
Functionals and basis functions:

l_{0} : v \mapsto \int_{f_{0}} v \cdot (s_{0} s_{1} - s_{0} - s_{1} + 1) {\hat{n}}_{0}

where

f_{0}

is the 0th face;

{\hat{n}}_{0}

is the normal to facet 0;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{0}

ϕ_{0} = (\begin{matrix} \frac{9 x (8 x y z - 4 x y - 6 x z + 3 x - 8 y z + 4 y + 6 z - 3)}{2} \\ \frac{9 y (8 x y z - 4 x y - 8 x z + 4 x - 6 y z + 3 y + 6 z - 3)}{2} \\ 36 x y z^{2} - 72 x y z + 36 x y - 27 x z^{2} + 51 x z - 24 x - 27 y z^{2} + 51 y z - 24 y + 21 z^{2} - 37 z + 16 \end{matrix})

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

l_{1} : v \mapsto \int_{f_{0}} v \cdot (s_{0} (1 - s_{1})) {\hat{n}}_{0}

where

f_{0}

is the 0th face;

{\hat{n}}_{0}

is the normal to facet 0;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{0}

ϕ_{1} = (\begin{matrix} \frac{9 x (- 8 x y z + 4 x y + 6 x z - 3 x + 8 y z - 4 y - 6 z + 3)}{2} \\ \frac{9 y (- 8 x y z + 4 x y + 8 x z - 4 x + 2 y z - y - 2 z + 1)}{2} \\ - 36 x y z^{2} + 72 x y z - 36 x y + 27 x z^{2} - 51 x z + 24 x + 9 y z^{2} - 21 y z + 12 y - 6 z^{2} + 14 z - 8 \end{matrix})

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

l_{2} : v \mapsto \int_{f_{0}} v \cdot (s_{1} (1 - s_{0})) {\hat{n}}_{0}

where

f_{0}

is the 0th face;

{\hat{n}}_{0}

is the normal to facet 0;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{0}

ϕ_{2} = (\begin{matrix} \frac{9 x (- 8 x y z + 4 x y + 2 x z - x + 8 y z - 4 y - 2 z + 1)}{2} \\ \frac{9 y (- 8 x y z + 4 x y + 8 x z - 4 x + 6 y z - 3 y - 6 z + 3)}{2} \\ - 36 x y z^{2} + 72 x y z - 36 x y + 9 x z^{2} - 21 x z + 12 x + 27 y z^{2} - 51 y z + 24 y - 6 z^{2} + 14 z - 8 \end{matrix})

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

l_{3} : v \mapsto \int_{f_{0}} v \cdot (s_{0} s_{1}) {\hat{n}}_{0}

where

f_{0}

is the 0th face;

{\hat{n}}_{0}

is the normal to facet 0;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{0}

ϕ_{3} = (\begin{matrix} \frac{9 x (8 x y z - 4 x y - 2 x z + x - 8 y z + 4 y + 2 z - 1)}{2} \\ \frac{9 y (8 x y z - 4 x y - 8 x z + 4 x - 2 y z + y + 2 z - 1)}{2} \\ 36 x y z^{2} - 72 x y z + 36 x y - 9 x z^{2} + 21 x z - 12 x - 9 y z^{2} + 21 y z - 12 y + 3 z^{2} - 7 z + 4 \end{matrix})

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

l_{4} : v \mapsto \int_{f_{1}} v \cdot (s_{0} s_{1} - s_{0} - s_{1} + 1) {\hat{n}}_{1}

where

f_{1}

is the 1st face;

{\hat{n}}_{1}

is the normal to facet 1;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{1}

ϕ_{4} = (\begin{matrix} \frac{9 x (- 8 x y z + 6 x y + 4 x z - 3 x + 8 y z - 6 y - 4 z + 3)}{2} \\ - 36 x y^{2} z + 27 x y^{2} + 72 x y z - 51 x y - 36 x z + 24 x + 27 y^{2} z - 21 y^{2} - 51 y z + 37 y + 24 z - 16 \\ \frac{9 z (- 8 x y z + 8 x y + 4 x z - 4 x + 6 y z - 6 y - 3 z + 3)}{2} \end{matrix})

This DOF is associated with face 1 of the reference element.

l_{5} : v \mapsto \int_{f_{1}} v \cdot (s_{0} (1 - s_{1})) {\hat{n}}_{1}

where

f_{1}

is the 1st face;

{\hat{n}}_{1}

is the normal to facet 1;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{1}

ϕ_{5} = (\begin{matrix} \frac{9 x (8 x y z - 6 x y - 4 x z + 3 x - 8 y z + 6 y + 4 z - 3)}{2} \\ 36 x y^{2} z - 27 x y^{2} - 72 x y z + 51 x y + 36 x z - 24 x - 9 y^{2} z + 6 y^{2} + 21 y z - 14 y - 12 z + 8 \\ \frac{9 z (8 x y z - 8 x y - 4 x z + 4 x - 2 y z + 2 y + z - 1)}{2} \end{matrix})

This DOF is associated with face 1 of the reference element.

l_{6} : v \mapsto \int_{f_{1}} v \cdot (s_{1} (1 - s_{0})) {\hat{n}}_{1}

where

f_{1}

is the 1st face;

{\hat{n}}_{1}

is the normal to facet 1;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{1}

ϕ_{6} = (\begin{matrix} \frac{9 x (8 x y z - 2 x y - 4 x z + x - 8 y z + 2 y + 4 z - 1)}{2} \\ 36 x y^{2} z - 9 x y^{2} - 72 x y z + 21 x y + 36 x z - 12 x - 27 y^{2} z + 6 y^{2} + 51 y z - 14 y - 24 z + 8 \\ \frac{9 z (8 x y z - 8 x y - 4 x z + 4 x - 6 y z + 6 y + 3 z - 3)}{2} \end{matrix})

This DOF is associated with face 1 of the reference element.

l_{7} : v \mapsto \int_{f_{1}} v \cdot (s_{0} s_{1}) {\hat{n}}_{1}

where

f_{1}

is the 1st face;

{\hat{n}}_{1}

is the normal to facet 1;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{1}

ϕ_{7} = (\begin{matrix} \frac{9 x (- 8 x y z + 2 x y + 4 x z - x + 8 y z - 2 y - 4 z + 1)}{2} \\ - 36 x y^{2} z + 9 x y^{2} + 72 x y z - 21 x y - 36 x z + 12 x + 9 y^{2} z - 3 y^{2} - 21 y z + 7 y + 12 z - 4 \\ \frac{9 z (- 8 x y z + 8 x y + 4 x z - 4 x + 2 y z - 2 y - z + 1)}{2} \end{matrix})

This DOF is associated with face 1 of the reference element.

l_{8} : v \mapsto \int_{f_{2}} v \cdot (s_{0} s_{1} - s_{0} - s_{1} + 1) {\hat{n}}_{2}

where

f_{2}

is the 2nd face;

{\hat{n}}_{2}

is the normal to facet 2;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{2}

ϕ_{8} = (\begin{matrix} 36 x^{2} y z - 27 x^{2} y - 27 x^{2} z + 21 x^{2} - 72 x y z + 51 x y + 51 x z - 37 x + 36 y z - 24 y - 24 z + 16 \\ \frac{9 y (8 x y z - 6 x y - 8 x z + 6 x - 4 y z + 3 y + 4 z - 3)}{2} \\ \frac{9 z (8 x y z - 8 x y - 6 x z + 6 x - 4 y z + 4 y + 3 z - 3)}{2} \end{matrix})

This DOF is associated with face 2 of the reference element.

l_{9} : v \mapsto \int_{f_{2}} v \cdot (s_{0} (1 - s_{1})) {\hat{n}}_{2}

where

f_{2}

is the 2nd face;

{\hat{n}}_{2}

is the normal to facet 2;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{2}

ϕ_{9} = (\begin{matrix} - 36 x^{2} y z + 27 x^{2} y + 9 x^{2} z - 6 x^{2} + 72 x y z - 51 x y - 21 x z + 14 x - 36 y z + 24 y + 12 z - 8 \\ \frac{9 y (- 8 x y z + 6 x y + 8 x z - 6 x + 4 y z - 3 y - 4 z + 3)}{2} \\ \frac{9 z (- 8 x y z + 8 x y + 2 x z - 2 x + 4 y z - 4 y - z + 1)}{2} \end{matrix})

This DOF is associated with face 2 of the reference element.

l_{10} : v \mapsto \int_{f_{2}} v \cdot (s_{1} (1 - s_{0})) {\hat{n}}_{2}

where

f_{2}

is the 2nd face;

{\hat{n}}_{2}

is the normal to facet 2;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{2}

ϕ_{10} = (\begin{matrix} - 36 x^{2} y z + 9 x^{2} y + 27 x^{2} z - 6 x^{2} + 72 x y z - 21 x y - 51 x z + 14 x - 36 y z + 12 y + 24 z - 8 \\ \frac{9 y (- 8 x y z + 2 x y + 8 x z - 2 x + 4 y z - y - 4 z + 1)}{2} \\ \frac{9 z (- 8 x y z + 8 x y + 6 x z - 6 x + 4 y z - 4 y - 3 z + 3)}{2} \end{matrix})

This DOF is associated with face 2 of the reference element.

l_{11} : v \mapsto \int_{f_{2}} v \cdot (s_{0} s_{1}) {\hat{n}}_{2}

where

f_{2}

is the 2nd face;

{\hat{n}}_{2}

is the normal to facet 2;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{2}

ϕ_{11} = (\begin{matrix} 36 x^{2} y z - 9 x^{2} y - 9 x^{2} z + 3 x^{2} - 72 x y z + 21 x y + 21 x z - 7 x + 36 y z - 12 y - 12 z + 4 \\ \frac{9 y (8 x y z - 2 x y - 8 x z + 2 x - 4 y z + y + 4 z - 1)}{2} \\ \frac{9 z (8 x y z - 8 x y - 2 x z + 2 x - 4 y z + 4 y + z - 1)}{2} \end{matrix})

This DOF is associated with face 2 of the reference element.

l_{12} : v \mapsto \int_{f_{3}} v \cdot (s_{0} s_{1} - s_{0} - s_{1} + 1) {\hat{n}}_{3}

where

f_{3}

is the 3rd face;

{\hat{n}}_{3}

is the normal to facet 3;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{3}

ϕ_{12} = (\begin{matrix} x (36 x y z - 27 x y - 27 x z + 21 x + 3 y + 3 z - 5) \\ \frac{9 y (8 x y z - 6 x y - 8 x z + 6 x - 4 y z + 3 y + 4 z - 3)}{2} \\ \frac{9 z (8 x y z - 8 x y - 6 x z + 6 x - 4 y z + 4 y + 3 z - 3)}{2} \end{matrix})

This DOF is associated with face 3 of the reference element.

l_{13} : v \mapsto \int_{f_{3}} v \cdot (s_{0} (1 - s_{1})) {\hat{n}}_{3}

where

f_{3}

is the 3rd face;

{\hat{n}}_{3}

is the normal to facet 3;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{3}

ϕ_{13} = (\begin{matrix} x (- 36 x y z + 27 x y + 9 x z - 6 x - 3 y + 3 z - 2) \\ \frac{9 y (- 8 x y z + 6 x y + 8 x z - 6 x + 4 y z - 3 y - 4 z + 3)}{2} \\ \frac{9 z (- 8 x y z + 8 x y + 2 x z - 2 x + 4 y z - 4 y - z + 1)}{2} \end{matrix})

This DOF is associated with face 3 of the reference element.

l_{14} : v \mapsto \int_{f_{3}} v \cdot (s_{1} (1 - s_{0})) {\hat{n}}_{3}

where

f_{3}

is the 3rd face;

{\hat{n}}_{3}

is the normal to facet 3;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{3}

ϕ_{14} = (\begin{matrix} x (- 36 x y z + 9 x y + 27 x z - 6 x + 3 y - 3 z - 2) \\ \frac{9 y (- 8 x y z + 2 x y + 8 x z - 2 x + 4 y z - y - 4 z + 1)}{2} \\ \frac{9 z (- 8 x y z + 8 x y + 6 x z - 6 x + 4 y z - 4 y - 3 z + 3)}{2} \end{matrix})

This DOF is associated with face 3 of the reference element.

l_{15} : v \mapsto \int_{f_{3}} v \cdot (s_{0} s_{1}) {\hat{n}}_{3}

where

f_{3}

is the 3rd face;

{\hat{n}}_{3}

is the normal to facet 3;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{3}

ϕ_{15} = (\begin{matrix} x (36 x y z - 9 x y - 9 x z + 3 x - 3 y - 3 z + 1) \\ \frac{9 y (8 x y z - 2 x y - 8 x z + 2 x - 4 y z + y + 4 z - 1)}{2} \\ \frac{9 z (8 x y z - 8 x y - 2 x z + 2 x - 4 y z + 4 y + z - 1)}{2} \end{matrix})

This DOF is associated with face 3 of the reference element.

l_{16} : v \mapsto \int_{f_{4}} v \cdot (s_{0} s_{1} - s_{0} - s_{1} + 1) {\hat{n}}_{4}

where

f_{4}

is the 4th face;

{\hat{n}}_{4}

is the normal to facet 4;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{4}

ϕ_{16} = (\begin{matrix} \frac{9 x (- 8 x y z + 6 x y + 4 x z - 3 x + 8 y z - 6 y - 4 z + 3)}{2} \\ y (- 36 x y z + 27 x y - 3 x + 27 y z - 21 y - 3 z + 5) \\ \frac{9 z (- 8 x y z + 8 x y + 4 x z - 4 x + 6 y z - 6 y - 3 z + 3)}{2} \end{matrix})

This DOF is associated with face 4 of the reference element.

l_{17} : v \mapsto \int_{f_{4}} v \cdot (s_{0} (1 - s_{1})) {\hat{n}}_{4}

where

f_{4}

is the 4th face;

{\hat{n}}_{4}

is the normal to facet 4;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{4}

ϕ_{17} = (\begin{matrix} \frac{9 x (8 x y z - 6 x y - 4 x z + 3 x - 8 y z + 6 y + 4 z - 3)}{2} \\ y (36 x y z - 27 x y + 3 x - 9 y z + 6 y - 3 z + 2) \\ \frac{9 z (8 x y z - 8 x y - 4 x z + 4 x - 2 y z + 2 y + z - 1)}{2} \end{matrix})

This DOF is associated with face 4 of the reference element.

l_{18} : v \mapsto \int_{f_{4}} v \cdot (s_{1} (1 - s_{0})) {\hat{n}}_{4}

where

f_{4}

is the 4th face;

{\hat{n}}_{4}

is the normal to facet 4;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{4}

ϕ_{18} = (\begin{matrix} \frac{9 x (8 x y z - 2 x y - 4 x z + x - 8 y z + 2 y + 4 z - 1)}{2} \\ y (36 x y z - 9 x y - 3 x - 27 y z + 6 y + 3 z + 2) \\ \frac{9 z (8 x y z - 8 x y - 4 x z + 4 x - 6 y z + 6 y + 3 z - 3)}{2} \end{matrix})

This DOF is associated with face 4 of the reference element.

l_{19} : v \mapsto \int_{f_{4}} v \cdot (s_{0} s_{1}) {\hat{n}}_{4}

where

f_{4}

is the 4th face;

{\hat{n}}_{4}

is the normal to facet 4;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{4}

ϕ_{19} = (\begin{matrix} \frac{9 x (- 8 x y z + 2 x y + 4 x z - x + 8 y z - 2 y - 4 z + 1)}{2} \\ y (- 36 x y z + 9 x y + 3 x + 9 y z - 3 y + 3 z - 1) \\ \frac{9 z (- 8 x y z + 8 x y + 4 x z - 4 x + 2 y z - 2 y - z + 1)}{2} \end{matrix})

This DOF is associated with face 4 of the reference element.

l_{20} : v \mapsto \int_{f_{5}} v \cdot (s_{0} s_{1} - s_{0} - s_{1} + 1) {\hat{n}}_{5}

where

f_{5}

is the 5th face;

{\hat{n}}_{5}

is the normal to facet 5;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{5}

ϕ_{20} = (\begin{matrix} \frac{9 x (8 x y z - 4 x y - 6 x z + 3 x - 8 y z + 4 y + 6 z - 3)}{2} \\ \frac{9 y (8 x y z - 4 x y - 8 x z + 4 x - 6 y z + 3 y + 6 z - 3)}{2} \\ z (36 x y z - 27 x z + 3 x - 27 y z + 3 y + 21 z - 5) \end{matrix})

This DOF is associated with face 5 of the reference element.

l_{21} : v \mapsto \int_{f_{5}} v \cdot (s_{0} (1 - s_{1})) {\hat{n}}_{5}

where

f_{5}

is the 5th face;

{\hat{n}}_{5}

is the normal to facet 5;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{5}

ϕ_{21} = (\begin{matrix} \frac{9 x (- 8 x y z + 4 x y + 6 x z - 3 x + 8 y z - 4 y - 6 z + 3)}{2} \\ \frac{9 y (- 8 x y z + 4 x y + 8 x z - 4 x + 2 y z - y - 2 z + 1)}{2} \\ z (- 36 x y z + 27 x z - 3 x + 9 y z + 3 y - 6 z - 2) \end{matrix})

This DOF is associated with face 5 of the reference element.

l_{22} : v \mapsto \int_{f_{5}} v \cdot (s_{1} (1 - s_{0})) {\hat{n}}_{5}

where

f_{5}

is the 5th face;

{\hat{n}}_{5}

is the normal to facet 5;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{5}

ϕ_{22} = (\begin{matrix} \frac{9 x (- 8 x y z + 4 x y + 2 x z - x + 8 y z - 4 y - 2 z + 1)}{2} \\ \frac{9 y (- 8 x y z + 4 x y + 8 x z - 4 x + 6 y z - 3 y - 6 z + 3)}{2} \\ z (- 36 x y z + 9 x z + 3 x + 27 y z - 3 y - 6 z - 2) \end{matrix})

This DOF is associated with face 5 of the reference element.

l_{23} : v \mapsto \int_{f_{5}} v \cdot (s_{0} s_{1}) {\hat{n}}_{5}

where

f_{5}

is the 5th face;

{\hat{n}}_{5}

is the normal to facet 5;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

f_{5}

ϕ_{23} = (\begin{matrix} \frac{9 x (8 x y z - 4 x y - 2 x z + x - 8 y z + 4 y + 2 z - 1)}{2} \\ \frac{9 y (8 x y z - 4 x y - 8 x z + 4 x - 2 y z + y + 2 z - 1)}{2} \\ z (36 x y z - 9 x z - 3 x - 9 y z - 3 y + 3 z + 1) \end{matrix})

This DOF is associated with face 5 of the reference element.

l_{24} : v \mapsto \int_{R} (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}) \cdot v

where

R

is the reference element.

ϕ_{24} = (\begin{matrix} 9 x (- 8 x y z + 4 x y + 6 x z - 3 x + 8 y z - 4 y - 6 z + 3) \\ 9 y (- 8 x y z + 4 x y + 8 x z - 4 x + 6 y z - 3 y - 6 z + 3) \\ 6 z (- 12 x y z + 12 x y + 9 x z - 9 x + 9 y z - 9 y - 7 z + 7) \end{matrix})

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

l_{25} : v \mapsto \int_{R} (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}) \cdot v

where

R

is the reference element.

ϕ_{25} = (\begin{matrix} 9 x (- 8 x y z + 6 x y + 4 x z - 3 x + 8 y z - 6 y - 4 z + 3) \\ 6 y (- 12 x y z + 9 x y + 12 x z - 9 x + 9 y z - 7 y - 9 z + 7) \\ 9 z (- 8 x y z + 8 x y + 4 x z - 4 x + 6 y z - 6 y - 3 z + 3) \end{matrix})

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

l_{26} : v \mapsto \int_{R} (\begin{matrix} 0 \\ s_{2} \\ s_{1} \end{matrix}) \cdot v

where

R

is the reference element;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

R

ϕ_{26} = (\begin{matrix} 36 x (4 x y z - 2 x y - 2 x z + x - 4 y z + 2 y + 2 z - 1) \\ 18 y (8 x y z - 4 x y - 8 x z + 4 x - 6 y z + 3 y + 6 z - 3) \\ 18 z (8 x y z - 8 x y - 4 x z + 4 x - 6 y z + 6 y + 3 z - 3) \end{matrix})

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

l_{27} : v \mapsto \int_{R} (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}) \cdot v

where

R

is the reference element.

ϕ_{27} = (\begin{matrix} 6 x (- 12 x y z + 9 x y + 9 x z - 7 x + 12 y z - 9 y - 9 z + 7) \\ 9 y (- 8 x y z + 6 x y + 8 x z - 6 x + 4 y z - 3 y - 4 z + 3) \\ 9 z (- 8 x y z + 8 x y + 6 x z - 6 x + 4 y z - 4 y - 3 z + 3) \end{matrix})

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

l_{28} : v \mapsto \int_{R} (\begin{matrix} s_{2} \\ 0 \\ s_{0} \end{matrix}) \cdot v

where

R

is the reference element;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

R

ϕ_{28} = (\begin{matrix} 18 x (8 x y z - 4 x y - 6 x z + 3 x - 8 y z + 4 y + 6 z - 3) \\ 36 y (4 x y z - 2 x y - 4 x z + 2 x - 2 y z + y + 2 z - 1) \\ 18 z (8 x y z - 8 x y - 6 x z + 6 x - 4 y z + 4 y + 3 z - 3) \end{matrix})

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

l_{29} : v \mapsto \int_{R} (\begin{matrix} s_{1} \\ s_{0} \\ 0 \end{matrix}) \cdot v

where

R

is the reference element;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

R

ϕ_{29} = (\begin{matrix} 18 x (8 x y z - 6 x y - 4 x z + 3 x - 8 y z + 6 y + 4 z - 3) \\ 18 y (8 x y z - 6 x y - 8 x z + 6 x - 4 y z + 3 y + 4 z - 3) \\ 36 z (4 x y z - 4 x y - 2 x z + 2 x - 2 y z + 2 y + z - 1) \end{matrix})

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

l_{30} : v \mapsto \int_{R} (\begin{matrix} s_{1} s_{2} \\ s_{0} s_{2} \\ s_{0} s_{1} \end{matrix}) \cdot v

where

R

is the reference element;
and

s_{0}, s_{1}, s_{2}

is a parametrisation of

R

ϕ_{30} = (\begin{matrix} 72 x (- 4 x y z + 2 x y + 2 x z - x + 4 y z - 2 y - 2 z + 1) \\ 72 y (- 4 x y z + 2 x y + 4 x z - 2 x + 2 y z - y - 2 z + 1) \\ 72 z (- 4 x y z + 4 x y + 2 x z - 2 x + 2 y z - 2 y - z + 1) \end{matrix})

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