an encyclopedia of finite element definitions

# Arnold–Boffi–Falk

 Abbreviated names ABF Orders $$0\leqslant k$$ Reference elements quadrilateral Polynomial set $$\mathcal{Z}^{(0)}_{k} \oplus \mathcal{Z}^{(1)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each edge: normal integral moments with an order $$k$$ Lagrange space On each face: integral moments with an order $$k$$ Nédélec (first kind) space,integral moments of the divergence with $$x^{k+1}y^q$$ for q=0,1,...,k, and integral moments of the divergence with $$x^qy^{k+1}$$ for q=0,1,...,k Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

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

## Examples

 quadrilateralorder 0 (click to view basis functions) quadrilateralorder 1 (click to view basis functions) quadrilateralorder 2 (click to view basis functions)
• Arnold, Douglas N., Boffi, Daniele, and Falk, Richard S. Quadrilateral H(div) Finite Elements, SIAM Journal on Numerical Analysis 42(5), 2429–2451, 2005. [DOI: 10.1137/S0036142903431924] [BibTeX]

## DefElement stats

 Element added 01 December 2021 Element last updated 16 September 2023