FIAT: enabling classical and modern macroelements

TL;DR

This paper introduces a comprehensive extension to the FIAT finite element library, enabling broad support for various macroelements that facilitate complex properties like divergence-free and $C^1$ continuity, integrated into the Firedrake framework.

Contribution

It presents a general approach for supporting diverse macroelements in finite element software, enhancing flexibility and efficiency over existing one-off solutions.

Findings

Supported a wide range of macroelements in FIAT

Integrated macroelement support into Firedrake

Numerical evaluation demonstrates effectiveness

Abstract

Many classical and modern finite element spaces are derived by dividing each computational cell into finer pieces. Such \emph{macroelements} frequently enable the enforcement of mathematically desirable properties such as divergence-free conditions or $C^1$ continuity in a simpler or more efficient manner than elements without the subdivision. Although a few modern software projects provide one-off support for particular macroelements, a general approach facilitating broad-based support has, until now, been lacking. In this work, we describe a major addition to the FIAT project to support a wide range of different macroelements. These enhancements have been integrated into the Firedrake code stack. We provide numerical evaluation of the new macroelement facility.