an encyclopedia of finite element definitions

# Huang–Zhang

 Alternative names $$Q_{k+1,k}\times Q_{k,k+1}$$ Abbreviated names HZ Orders $$2\leqslant k$$ Reference elements quadrilateral Polynomial set $$\mathcal{Z}^{(13)}_{k} \oplus \mathcal{Z}^{(14)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k-1$$ Lagrange space, and tangent integral moments with an order $$k-2$$ Lagrange space On the interior of the reference element: integral moments with $$\left\{\left(\begin{array}{c}x^iy^j\\0\end{array}\right)\middle|i\in\{0,1,...,k-1\}, j\in\{0,1,...,k-2\}\right\}\cup\left\{\left(\begin{array}{c}x^iy^j\\0\end{array}\right)\middle|i\in\{0,1,...,k-2\}, j\in\{0,1,...,k-1\}\right\}$$ Number of DOFs quadrilateral: $$2k(k+1)$$ (A046092) Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem "HZ"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations.

## Examples

 quadrilateralorder 2 (click to view basis functions) quadrilateralorder 3 (click to view basis functions)
• 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 16 September 2023