Transformations for Piola-mapped elements

TL;DR

This paper develops a new transformation method for Piola-mapped elements, enabling Arnold-Winther and Mardal-Tai-Winther elements to be used in standard finite element software like Firedrake, with verified numerical results.

Contribution

It introduces a novel transformation theory for Piola-mapped elements, allowing their implementation in common finite element software.

Findings

Successfully implemented Arnold-Winther elements in Firedrake.

Verified the correctness of the transformation through numerical tests.

Enabled the use of advanced elements for elasticity and fluid flow in standard software.

Abstract

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