Code generation for generally mapped finite elements

TL;DR

This paper introduces a method to implement and evaluate general finite element transformations in FInAT and Firedrake, enabling the use of classical elements like Argyris and Bell for complex PDE problems.

Contribution

It extends finite element software to support a wide range of classical elements, improving solution smoothness and performance for higher-order PDEs.

Findings

New elements provide smooth solutions with mild cost increase for second-order problems.

Significant performance improvements for fourth-order problems over traditional methods.

Successfully applied to nonlinear Cahn-Hilliard and biharmonic eigenvalue problems.

Abstract

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite element transformations in FInAT and hence into the Firedrake finite element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly-enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using $C^1$ discretizations.