New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

TL;DR

This paper introduces new symmetric cubature rules on the triangle that improve existing rules by using minimal orthonormal bases, leading to better performance and fewer points needed.

Contribution

It presents novel symmetric cubature rules on the triangle derived from minimal orthonormal bases, enhancing efficiency and quality over previous methods.

Findings

New rules outperform existing ones in point count and quality

Implementation of minimal orthonormal bases improves algorithm performance

Benchmark examples demonstrate significant improvements

Abstract

Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with better characteristics. There is therefore clear interest in searching for better cubature rules. Here we present a number of new cubature rules on the triangle, exhibiting full or rotational symmetry, that improve on those available in the literature either in terms of number of points or in terms of quality. These rules were obtained by determining and implementing minimal orthonormal polynomial bases that can express the symmetries of the cubature rules. As shown in specific benchmark examples, this results in significantly better performance of the employed algorithm.