an encyclopedia of finite element definitions

# Arnold–Winther

 Alternative names conforming Arnold–Winther Orders $$3\leqslant k$$ Reference elements triangle Polynomial set $$\mathcal{Z}^{(2)}_{k-1} \oplus \mathcal{Z}^{(3)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations of three components On each edge: integral moments of normal-normal and normal-tangent inner products with an order $$k-2$$ Lagrange space On each face: integral moments of three components with an order $$k-3$$ Lagrange space, and integral moments of tensor dot product with $$\frac{\partial}{\partial(x, y)}x^2y^2(1-x-y)^2f$$ for each order $$k-4$$ polynomial $$f$$ in an order $$k-4$$ Lagrange space Number of DOFs triangle: $$(3k^2+5k+6)/2$$ Categories Matrix-valued elements

## Implementations

 FIAT FIAT.ArnoldWinther↓ Show FIAT examples ↓ This implementation is correct for all the examples below. Symfem "AW"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "AWc"↓ Show (legacy) UFL examples ↓

## Examples

 triangleorder 3 (click to view basis functions) triangleorder 4 (click to view basis functions)

## References

• Arnold, Douglas N. and Winther, Ragnar. Mixed finite elements for elasticity, Numerische Mathematik 92(3), 401–419, 2002. [DOI: 10.1007/s002110100348] [BibTeX]

## DefElement stats

 Element added 10 February 2021 Element last updated 16 September 2023