Analytical Formulae for Two of A. H. Stroud's Quadrature Rules

TL;DR

This paper provides explicit analytical formulas for two fifth-order quadrature rules on the 3-cube, enhancing understanding and practical application in finite element methods.

Contribution

It derives and documents analytical formulas for Stroud's quadrature rules, which were previously not well detailed, improving their usability in computational applications.

Findings

Explicit formulas for points and weights are provided.

The rules are suitable for finite element calculations.

Enhanced understanding of Stroud's quadrature rules.

Abstract

Analytical formulae for the points and weights of two fifth-order quadrature rules for C_3, the 3-cube, are given. The rules, originally formulated by A. H. Stroud in 1967, are discussed in greater detail in terms of both the setup of the basic equations and the method of obtaining their solutions analytically. The primary purpose of this paper is to better document what we feel is a particularly practical quadrature rule (e.g. in finite element calculations) and one for which we felt comprehensive information was scarce.