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Degree 0 Gopalakrishnan–Lederer–Schöberl on a tetrahedron
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- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{ccc}\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{ccc}\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 1\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle - \frac{2 \sqrt{3}}{3}\\\displaystyle \frac{2 \sqrt{3}}{3}\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)\)
where \(f_{0}\) is the 0th face.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{2}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle - \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{ccc}\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{2}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle - \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle - \frac{2 \sqrt{3}}{3}\\\displaystyle 0\\\displaystyle \frac{2 \sqrt{3}}{3}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)\)
where \(f_{0}\) is the 0th face.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{2}{3}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{ccc}\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{2}{3}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
where \(f_{1}\) is the 1st face.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle - \frac{2}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{ccc}\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle - \frac{2}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
where \(f_{1}\) is the 1st face.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{1}{3}&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle - \frac{2}{3}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{ccc}\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{1}{3}&\displaystyle 0\\\displaystyle 1&\displaystyle 0&\displaystyle - \frac{2}{3}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 2\\\displaystyle 0\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(f_{2}\) is the 2nd face.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle \frac{2}{3}&\displaystyle -1&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle - \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{ccc}\displaystyle \frac{2}{3}&\displaystyle -1&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle - \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(f_{2}\) is the 2nd face.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle -1&\displaystyle \frac{2}{3}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{ccc}\displaystyle - \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle -1&\displaystyle \frac{2}{3}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 2\\\displaystyle 0\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(f_{3}\) is the 3rd face.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{ccc}\displaystyle - \frac{2}{3}&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{ccc}\displaystyle - \frac{2}{3}&\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle \frac{1}{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{V}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(f_{3}\) is the 3rd face.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{ccc}\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{2}{3}&\displaystyle 1\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{ccc}\displaystyle \frac{1}{3}&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{2}{3}&\displaystyle 1\\\displaystyle 0&\displaystyle 0&\displaystyle \frac{1}{3}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{ccc}\displaystyle 2&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 2&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 2\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{ccc}\displaystyle 2&\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 2&\displaystyle 0\\\displaystyle 0&\displaystyle 0&\displaystyle 2\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.