Many classical and modern finite element spaces are derived by dividing each computational cell into finer pieces. Such 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.

FIAT (the FInite element Automatic Tabulator) provides a powerful Python library for the generation and evaluation of finite element basis functions on a reference element. This release paper describes recent improvements to FIAT aimed at improving its run time and the accuracy and efficiency of code generated using FIAT-provided information. In the first category, we have greatly streamlined the implementation of orthogonal polynomials out of which finite element bases are built. The second category comprises several more advances. For one, we have built an interface to the recursivenodes package to enable more accurate Lagrange bases at high order. We have also implemented integral-type degrees of freedom for \(H(\operatorname{div})\) and \(H(\operatorname{curl})\) elements, which match the mathematical definitions of the elements more closely and also avoid loss of accuracy in interpolation. More fundamentally, we have included families of simplicial quadrature rules that require many fewer quadrature points than the Stroud rules previously used in FIAT. Finally, FIAT now provides support for fast diagonalization methods, which enable fast solution algorithms at very high order. In each case, we describe the new features in FIAT and illustrate some of the gains obtained through simple numerical tests.