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Huang–Zhang
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| Alternative names | \(Q_{k+1,k}\times Q_{k,k+1}\) |
| Abbreviated names | HZ |
| Degrees | \(1\leqslant k\) where \(k\) is the Polynomial subdegree |
| Polynomial subdegree | \(k\) |
| Polynomial superdegree | \(2k+1\) |
| Lagrange subdegree | \(k\) |
| Lagrange superdegree | \(k+1\) |
| Reference elements | quadrilateral |
| Polynomial set | \(\mathcal{Z}^{(15)}_{k+1} \oplus \mathcal{Z}^{(16)}_{k+1}\) ↓ Show polynomial set definitions ↓ |
| DOFs | On each facet: normal integral moments with a degree \(k\) Lagrange space, and tangent integral moments with a degree \(k-1\) Lagrange space On the interior of the reference element: integral moments with \(\left\{\left(\begin{array}{c}x^iy^j\\0\end{array}\right)\middle|i\in\{0,1,...,k\}, j\in\{0,1,...,k-1\}\right\}\cup\left\{\left(\begin{array}{c}x^iy^j\\0\end{array}\right)\middle|i\in\{0,1,...,k-1\}, j\in\{0,1,...,k\}\right\}\) |
| Number of DOFs | quadrilateral: \(2(k+1)(k+2)\) (A046092) |
| Mapping | contravariant Piola |
| continuity | Components normal to facets are continuous |
| Categories | Vector-valued elements, H(div) conforming elements |
Implementations
This element is implemented in Symfem .↓ Show implementation detail ↓Examples
| quadrilateral degree 1 |  (click to view basis functions) |
| quadrilateral degree 2 |  (click to view basis functions) |
References
- Zhang, Shangyou. A family of \(Q_{k+1,k}\times Q+{k,k+1}\) divergence-free finite elements on rectangular grids, SIAM journal on numerical analysis 47(3), 2090–2107, 2009. [DOI: 10.1137/080728949] [BibTeX]
- Huang, Yunqing and Zhang, Shangyou. A lowest order divergence-free finite element on rectangular grids, Frontiers of mathematics in China 6, 253–270, 2011. [DOI: 10.1007/s11464-011-0094-0] [BibTeX]
DefElement stats
| Element added | 09 December 2022 |
| Element last updated | 31 March 2025 |