Symmetric Triangle Quadrature Rules for Arbitrary Functions

TL;DR

This paper introduces two methods for symmetric triangle quadrature rules capable of integrating arbitrary functions, especially singular integrands, achieving significantly improved accuracy over traditional polynomial-based rules.

Contribution

It develops two novel approaches for symmetric triangle quadrature rules that effectively handle non-polynomial and singular functions, expanding the applicability of quadrature methods.

Findings

Both approaches achieve relative errors two orders of magnitude lower than polynomial rules.

The first approach is efficient for moderate point counts, while the second is better suited for larger point sets.

Demonstrated effectiveness on singular integrands.

Abstract

Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of points, though it is less efficient than the first approach. We demonstrate the effectiveness of both approaches on singular integrands, which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules.