Wu–Xu
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Orders | interval: \(k=2\)
triangle: \(k=3\)
tetrahedron: \(k=4\) |
Reference elements | interval, triangle, tetrahedron |
Polynomial set | \(\mathcal{P}_{k-1} \oplus \mathcal{Z}^{(52)}_{k}\) (interval)
\(\mathcal{P}_{k-1} \oplus \mathcal{Z}^{(53)}_{k}\) (triangle)
\(\mathcal{P}_{k-1} \oplus \mathcal{Z}^{(54)}_{k}\) (tetrahedron)
↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑\(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i\leqslant k\right\}\)
\(\mathcal{Z}^{(52)}_k=\left\{x(1-x)p\middle|p\in\mathcal{P}_{k-1}\setminus\{1\}\right\}\)
\(\mathcal{Z}^{(53)}_k=\left\{xy(1-x-y)p\middle|p\in\mathcal{P}_{k-1}\setminus\{1\}\right\}\)
\(\mathcal{Z}^{(54)}_k=\left\{xyz(1-x-y-z)p\middle|p\in\mathcal{P}_{k-1}\setminus\{1\}\right\}\) |
DOFs | On each vertex: point evaluations
On each edge: integrals of normal derivatives
On each face: integrals of normal derivatives
On each volume: integrals of normal derivatives |
Number of DOFs | interval: \(4\) triangle: \(12\) tetrahedron: \(38\) |
Mapping | identity |
continuity | Function values are continuous. |
Categories | Scalar-valued elements |
Implementations
Symfem | "Wu-Xu" ↓ 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 Wu-Xu order 2 on a interval element = symfem.create_element("interval", "Wu-Xu", 2)
# Create Wu-Xu order 3 on a triangle element = symfem.create_element("triangle", "Wu-Xu", 3)
# Create Wu-Xu order 4 on a tetrahedron element = symfem.create_element("tetrahedron", "Wu-Xu", 4) |
Examples
References
- Wu, Shuonan and Xu, Jinchao. Nonconforming finite element spaces for 2mth order partial differential equations on Rn simplical grids when m=n+1, Mathematics of computation 88, 531–551, 2019. [DOI: 10.1090/mcom/3361] [BibTeX]
DefElement stats
Element added | 08 June 2021 |
Element last updated | 02 August 2022 |