an encyclopedia of finite element definitions
Degree 2 enriched vector Galerkin on a tetrahedron
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- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
- Basis functions:
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 x^{2} + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle x \left(2 x - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(2 y - 1\right)\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle z \left(2 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(2 z - 1\right)\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(2 z - 1\right)\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 4 y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y z\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y z\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 4 x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x z\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x z\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 4 x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 4 z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z \left(- x - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(- x - y - z + 1\right)\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 4 y \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- x - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y \left(- x - y - z + 1\right)\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 4 x \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x \left(- x - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x \left(- x - y - z + 1\right)\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle x - \frac{1}{4}\\\displaystyle y - \frac{1}{4}\\\displaystyle z - \frac{1}{4}\end{array}\right)\)