an encyclopedia of finite element definitions

an encyclopedia of finite element definitions

- \(R\) 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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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