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Degree 1 Crouzeix–Raviart on a tetrahedron
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- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference cell is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(y\), \(z\)
- \(\mathcal{L}=\{l_0,...,l_{3}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{0} = 3 x + 3 y + 3 z - 2\)
This DOF is associated with face 0 of the reference cell.
\(\displaystyle \phi_{0} = 3 x + 3 y + 3 z - 2\)
This DOF is associated with face 0 of the reference cell.
\(\displaystyle l_{1}:v\mapsto v(0,\tfrac{1}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{1} = 1 - 3 x\)
This DOF is associated with face 1 of the reference cell.
\(\displaystyle \phi_{1} = 1 - 3 x\)
This DOF is associated with face 1 of the reference cell.
\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{3},0,\tfrac{1}{3})\)
\(\displaystyle \phi_{2} = 1 - 3 y\)
This DOF is associated with face 2 of the reference cell.
\(\displaystyle \phi_{2} = 1 - 3 y\)
This DOF is associated with face 2 of the reference cell.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3},0)\)
\(\displaystyle \phi_{3} = 1 - 3 z\)
This DOF is associated with face 3 of the reference cell.
\(\displaystyle \phi_{3} = 1 - 3 z\)
This DOF is associated with face 3 of the reference cell.