an encyclopedia of finite element definitions

# Degree 2 Lagrange on a hexahedron

◀ Back to Lagrange definition page In this example:
• $$R$$ 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:
$$\displaystyle l_{0}:v\mapsto v(0,0,0)$$

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

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.
$$\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{8} = \sqrt{3} x \left(x y z - x y - x z + x - y z + y + z - 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{9}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{9} = \sqrt{3} y \left(x y z - x y - x z + x - y z + y + z - 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{10} = \sqrt{3} z \left(x y z - x y - x z + x - y z + y + z - 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{3}$$ is the 3rd edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{11} = \sqrt{3} x y \left(- y z + y + z - 1\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{e_{4}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{4}$$ is the 4th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{4}$$.

$$\displaystyle \phi_{12} = \sqrt{3} x z \left(- y z + y + z - 1\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{e_{5}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{5}$$ is the 5th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{5}$$.

$$\displaystyle \phi_{13} = \sqrt{3} x y \left(- x z + x + z - 1\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{e_{6}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{6}$$ is the 6th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{6}$$.

$$\displaystyle \phi_{14} = \sqrt{3} y z \left(- x z + x + z - 1\right)$$

This DOF is associated with edge 6 of the reference element.
$$\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{e_{7}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{7}$$ is the 7th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{7}$$.

$$\displaystyle \phi_{15} = \sqrt{3} x y z \left(z - 1\right)$$

This DOF is associated with edge 7 of the reference element.
$$\displaystyle l_{16}:\mathbf{v}\mapsto\displaystyle\int_{e_{8}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{8}$$ is the 8th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{8}$$.

$$\displaystyle \phi_{16} = \sqrt{3} x z \left(- x y + x + y - 1\right)$$

This DOF is associated with edge 8 of the reference element.
$$\displaystyle l_{17}:\mathbf{v}\mapsto\displaystyle\int_{e_{9}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{9}$$ is the 9th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{9}$$.

$$\displaystyle \phi_{17} = \sqrt{3} y z \left(- x y + x + y - 1\right)$$

This DOF is associated with edge 9 of the reference element.
$$\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{e_{10}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{10}$$ is the 10th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{10}$$.

$$\displaystyle \phi_{18} = \sqrt{3} x y z \left(y - 1\right)$$

This DOF is associated with edge 10 of the reference element.
$$\displaystyle l_{19}:\mathbf{v}\mapsto\displaystyle\int_{e_{11}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{11}$$ is the 11th edge;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{11}$$.

$$\displaystyle \phi_{19} = \sqrt{3} x y z \left(x - 1\right)$$

This DOF is associated with edge 11 of the reference element.
$$\displaystyle l_{20}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$f_{0}$$ is the 0th face;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{0}$$.

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{21}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$f_{1}$$ is the 1st face;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{1}$$.

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{22}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$f_{2}$$ is the 2nd face;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{2}$$.

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{23}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$f_{3}$$ is the 3rd face;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{3}$$.

$$\displaystyle \phi_{23} = 3 x y z \left(y z - y - z + 1\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{24}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$f_{4}$$ is the 4th face;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{4}$$.

$$\displaystyle \phi_{24} = 3 x y z \left(x z - x - z + 1\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{25}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$f_{5}$$ is the 5th face;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{5}$$.

$$\displaystyle \phi_{25} = 3 x y z \left(x y - x - y + 1\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{26}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \cdot \left(648000 s_{0}^{2} s_{1}^{2} s_{2}^{2} - 648000 s_{0}^{2} s_{1}^{2} s_{2} + 108000 s_{0}^{2} s_{1}^{2} - 648000 s_{0}^{2} s_{1} s_{2}^{2} + 648000 s_{0}^{2} s_{1} s_{2} - 108000 s_{0}^{2} s_{1} + 108000 s_{0}^{2} s_{2}^{2} - 108000 s_{0}^{2} s_{2} + 18000 s_{0}^{2} - 648000 s_{0} s_{1}^{2} s_{2}^{2} + 648000 s_{0} s_{1}^{2} s_{2} - 108000 s_{0} s_{1}^{2} + 648000 s_{0} s_{1} s_{2}^{2} - 648000 s_{0} s_{1} s_{2} + 108000 s_{0} s_{1} - 108000 s_{0} s_{2}^{2} + 108000 s_{0} s_{2} - 18000 s_{0} + 108000 s_{1}^{2} s_{2}^{2} - 108000 s_{1}^{2} s_{2} + 18000 s_{1}^{2} - 108000 s_{1} s_{2}^{2} + 108000 s_{1} s_{2} - 18000 s_{1} + 18000 s_{2}^{2} - 18000 s_{2} + 3000\right))v$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{26} = 3 \sqrt{3} x y z \left(x y z - x y - x z + x - y z + y + z - 1\right)$$

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