Transformations for Piola-mapped elements

The Arnold–Winther element successfully discretizes the Hellinger–Reissner variational formulation of linear elasticity; its development was one key early breakthroughs finite exterior calculus. Despite great utility, it is not available in standard software, because degrees freedom are preserved under Piola push-forward. In this work we apply novel transformation theory recently developed by Kirby [SMAI J. Comput. Math., 4:197–224, 2018] to devise correct map for transforming basis on a reference cell generic physical triangle. This enables use elements, both conforming and nonconforming, widely-used Firedrake composing with advanced symbolic code generation geometric multigrid functionality. Similar results also enable Mardal–Tai–Winther incompressible fluid flow. We present numerical verifying correctness our theory.