# Piola-mapped finite elements in Firedrake for linear elasticity and Stokes flow

**Francis Aznaran** (University of Oxford, 🇬🇧)

Patrick Farrell

(University of Oxford, 🇬🇧)

Robert Kirby

(Baylor University, 🇺🇸)

Tuesday session 2 (Zoom) (15:00–16:30 GMT)

You can cite this talk by using the following BibTeΧ:

@incollection{fenics2021-aznaran,

title = {Piola-mapped finite elements in Firedrake for linear elasticity and Stokes flow},

author = {Francis Aznaran and Patrick Farrell and Robert Kirby},

year = {2021},

url = {http://mscroggs.github.io/fenics2021/talks/aznaran.html},

booktitle = {Proceedings of FEniCS 2021, online, 22--26 March},

editor = {Igor Baratta and J{\o}rgen S. Dokken and Chris Richarson and Matthew W. Scroggs},

doi = {10.6084/m9.figshare.14495295},

pages = {258--301}

}

Hide citation infoMany finite element spaces are not preserved by the standard pullback to the reference cell. Robust implementation therefore requires studying the relation between degrees of freedom under pushforward, in order to obtain the correct bases on a generic physical triangle [1]. In this work, we extend this transformation theory to vector- and tensor-valued elements mapped by the contravariant Piola transform. We apply this theory, and describe its efficient implementation in Firedrake, for the the Mardal–Tai–Winther elements discretizing \(H(\mathrm{div})\) for Stokes–Darcy flow, and the Arnold–Winther elements discretizing \(H(\operatorname{div};\mathbb{S})\) for the stress-displacement formulation of linear elasticity.

In particular, the Arnold–Winther elements were the first to stably enforce exact symmetry of the Cauchy stress tensor; we demonstrate how they may be efficiently mapped, while the few prior implementations are either custom-made for specific numerical experiments, or require the explicit element-by-element construction of the basis. Our novel implementation of these exotic elements composes inexpensively and automatically with the rest of the Firedrake code stack; numerical results are presented.

## References

- [1] R. C. Kirby. A general approach to transforming finite elements,
*The SMAI journal of computational mathematics* 4, 197–224, 2018. [DOI: 10.5802/smai-jcm.33]