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

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. These rules were obtained by determining and implementing minimal orthonormal polynomial bases that can express the symmetries of the cubature rules. As shown in a specific benchmark example, this results in significantly better performance of the employed algorithm. This is the first time that a large number of rotationally symmetric rules are obtained, with these rules being in most cases of good quality and having less points than the best available fully symmetric rules.