On the Identification of Symmetric Quadrature Rules for Finite Element Methods
For researchers using finite element methods, this provides a systematic way to generate better quadrature rules, though the improvement is incremental over known tabulated rules.
The paper presents a methodology for automatically identifying symmetric quadrature rules with positive weights and interior points for various finite element domains, producing many new and improved rules compared to existing literature.
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.