On the Identification of Symmetric Quadrature Rules for Finite Element Methods

In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying an ensemble of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.