Bernardi–Raugel
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Abbreviated names | BR |
Orders | \(k=1\) |
Reference elements | triangle, tetrahedron |
Polynomial set | \(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(4)}_{k}\) (triangle)
\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(5)}_{k}\) (tetrahedron, \(k=1\))
\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(6)}_{k}\) (tetrahedron, \(k=2\))
↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑\(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i\leqslant k\right\}\)
\(\mathcal{Z}^{(4)}_k=\left\{\left(\begin{array}{c}xy\\xy\end{array}\right),\left(\begin{array}{c}y(1-x-y)\\0\end{array}\right),\left(\begin{array}{c}0\\x(1-x-y)\end{array}\right)\right\}\)
\(\mathcal{Z}^{(5)}_k=\left\{\left(\begin{array}{c}xyz\\xyz\\xyz\end{array}\right),\left(\begin{array}{c}yz(1-x-y-z)\\0\\0\end{array}\right),\left(\begin{array}{c}0\\xz(1-x-y-z)\\0\end{array}\right),\left(\begin{array}{c}0\\0\\xy(1-x-y-z)\end{array}\right)\right\}\)
\(\mathcal{Z}^{(6)}_k=\left\{\left(\begin{array}{c}xyz\\xyz\\xyz\end{array}\right),\left(\begin{array}{c}yz(1-x-y-z)\\0\\0\end{array}\right),\left(\begin{array}{c}0\\xz(1-x-y-z)\\0\end{array}\right),\left(\begin{array}{c}0\\0\\xy(1-x-y-z)\end{array}\right),\left(\begin{array}{c}xyz(1-x-y-z)\\0\\0\end{array}\right),\left(\begin{array}{c}0\\xyz(1-x-y-z)\\0\end{array}\right),\left(\begin{array}{c}0\\0\\xyz(1-x-y-z)\end{array}\right)\right\}\) |
DOFs | On each edge: (if \(k>1\)) point evaluations in tangential directions at midpoints
On each facet: point evaluations in normal directions at vertices, normal integral moments with an order \(k-1\) Lagrange space, and (if \(k>1\)) point evaluations in normal directions at midpoints of edges
On the interior of the reference element: integral moments of the divergence with an order \(0\) vector Lagrange space |
Number of DOFs | triangle: \(9\) tetrahedron: \(\begin{cases}16&k=1\\37&k=2\end{cases}\) |
Mapping | contravariant Piola |
continuity | Components normal to facets are continuous |
Categories | Vector-valued elements, H(div) conforming elements |
Implementations
Symfem | "Bernardi-Raugel" ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑Before trying this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem
# Create Bernardi-Raugel order 1 on a triangle element = symfem.create_element("triangle", "Bernardi-Raugel", 1)
# Create Bernardi-Raugel order 1 on a tetrahedron element = symfem.create_element("tetrahedron", "Bernardi-Raugel", 1)
# Create Bernardi-Raugel order 2 on a tetrahedron element = symfem.create_element("tetrahedron", "Bernardi-Raugel", 2) |
Examples
References
DefElement stats
Element added | 19 April 2021 |
Element last updated | 02 August 2022 |