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Bernardi–Raugel
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| Abbreviated names | BR |
| Orders | \(k=1\) |
| Reference elements | triangle, tetrahedron |
| Polynomial set | \(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(7)}_{k}\) (triangle) \(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(8)}_{k}\) (tetrahedron) ↓ Show polynomial set definitions ↓ |
| DOFs | On each vertex: point evaluations in $d$ directions On each facet: normal integral moments with an order \(k-1\) Lagrange space |
| Number of DOFs | triangle: \(9\) tetrahedron: \(16\) |
| Categories | Vector-valued elements, H(div) conforming elements |
Implementations
| Symfem string | "Bernardi-Raugel"↓ Show Symfem examples ↓ |
Examples
triangleorder 1tetrahedron
order 1
- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2} x y}{2}\\\displaystyle - \frac{\sqrt{2} x y}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y \left(x + y - 1\right)\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(- x - y + 1\right)\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}-1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 3 x y + x - 3 y^{2} + 4 y - 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 3 x y + x - 3 y^{2} + 4 y - 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x^{2} + 3 x y - 4 x - y + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x^{2} + 3 x y - 4 x - y + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}-1\\-1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{x \left(3 y - 2\right)}{2}\\\displaystyle \frac{3 x y}{2}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{x \left(3 y - 2\right)}{2}\\\displaystyle \frac{3 x y}{2}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle x \left(3 x + 3 y - 2\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle x \left(3 x + 3 y - 2\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}-1\\-1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{3 x y}{2}\\\displaystyle \frac{y \left(3 x - 2\right)}{2}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{3 x y}{2}\\\displaystyle \frac{y \left(3 x - 2\right)}{2}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}-1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle y \left(- 3 x - 3 y + 2\right)\\\displaystyle y\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle y \left(- 3 x - 3 y + 2\right)\\\displaystyle y\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - 3 x y\\\displaystyle - 3 x y\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - 3 x y\\\displaystyle - 3 x y\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 y \left(x + y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 y \left(x + y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(- x - y - z + 1\right)\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{15}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 20 x y z - x + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 20 x y z - x + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 20 x^{2} z - 20 x y z - 20 x z^{2} + 20 x z + x + y + z - 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 20 x^{2} z - 20 x y z - 20 x z^{2} + 20 x z + x + y + z - 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 20 x^{2} y + 20 x y^{2} + 20 x y z - 20 x y - x - y - z + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 20 x^{2} y + 20 x y^{2} + 20 x y z - 20 x y - x - y - z + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 y z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 y z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle x\\\displaystyle x \left(- 20 x z - 20 y z - 20 z^{2} + 20 z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle x\\\displaystyle x \left(- 20 x z - 20 y z - 20 z^{2} + 20 z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle 0\\\displaystyle x \left(20 x y + 20 y^{2} + 20 y z - 20 y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle 0\\\displaystyle x \left(20 x y + 20 y^{2} + 20 y z - 20 y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{y \left(- 20 x z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{y \left(- 20 x z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle y \left(20 x z + 20 y z + 20 z^{2} - 20 z + 1\right)\\\displaystyle - y\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle y \left(20 x z + 20 y z + 20 z^{2} - 20 z + 1\right)\\\displaystyle - y\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - y\\\displaystyle y \left(20 x^{2} + 20 x y + 20 x z - 20 x + 1\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - y\\\displaystyle y \left(20 x^{2} + 20 x y + 20 x z - 20 x + 1\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{z \left(- 20 x y + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle \frac{z \left(- 20 x y + 3\right)}{3}\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle z \left(20 x y + 20 y^{2} + 20 y z - 20 y + 1\right)\\\displaystyle 0\\\displaystyle - z\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle z \left(20 x y + 20 y^{2} + 20 y z - 20 y + 1\right)\\\displaystyle 0\\\displaystyle - z\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(- 20 x^{2} - 20 x y - 20 x z + 20 x - 1\right)\\\displaystyle z\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(- 20 x^{2} - 20 x y - 20 x z + 20 x - 1\right)\\\displaystyle z\end{array}\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 40 x y z\\\displaystyle 40 x y z\\\displaystyle 40 x y z\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 40 x y z\\\displaystyle 40 x y z\\\displaystyle 40 x y z\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 120 y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 120 y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 120 x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 120 x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 120 x y \left(- x - y - z + 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 120 x y \left(- x - y - z + 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
References
- Bernardi, C. and Raugel, G. Analysis of some finite elements for the Stokes problem, Mathematics of Computation 44, 71–79, 1985. [DOI: 10.1090/S0025-5718-1985-0771031-7] [BibTeX]