FIAT: improving performance and accuracy for high-order finite elements

TL;DR

FIAT is a Python library for finite element basis functions that has been improved to enhance run time, accuracy, and efficiency through better polynomial implementation, high-order bases, integral degrees of freedom, efficient quadrature, and fast diagonalization methods.

Contribution

The paper introduces recent enhancements to FIAT, including optimized polynomial implementation, high-order basis accuracy, integral degrees of freedom, efficient quadrature rules, and support for fast diagonalization.

Findings

Streamlined orthogonal polynomial implementation.

More accurate high-order Lagrange bases.

Reduced quadrature points with new simplicial rules.

Abstract

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 $\texttt{recursivenodes}$ package to enable more accurate Lagrange bases at high order. We have also implemented integral-type degrees of freedom for $H(\mathrm{div})$ and $H(\mathrm{curl})$ 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.