Several new quadrature formulas for polynomial integration in the triangle

TL;DR

This paper introduces several new quadrature formulas for polynomial integration over triangles, achieving higher degrees of exactness with positive weights and no points outside the domain, advancing numerical integration methods.

Contribution

The paper develops new quadrature formulas for triangles up to degree 25, with positive weights and improved accuracy over previous methods, using a novel cardinal function algorithm.

Findings

Seven formulas outperform existing ones in accuracy

All formulas have positive weights and are confined within the triangle

Quadrature formulas are computed for degrees up to 25

Abstract

We present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that the number of quadrature points $N$ be equal to the dimension of a lower dimensional polynomial space. Quadrature forumulas are presented for up to degree $d=25$, all which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results.