Very high-order symmetric positive-interior quadrature rules on triangles and tetrahedra

TL;DR

This paper introduces highly efficient, symmetric positive-interior quadrature rules of very high degrees for triangles and tetrahedra, improving accuracy and efficiency in numerical integration over these domains.

Contribution

It develops novel quadrature rules with positive weights and interior nodes up to degree 84 on triangles and 40 on tetrahedra, using tensor-product initial guesses and node elimination techniques.

Findings

Quadrature rules are over 95% efficient on triangles.

Quadrature rules are over 80% efficient on tetrahedra.

Numerical examples confirm high accuracy of the rules.

Abstract

We present novel fully-symmetric quadrature rules with positive weights and strictly interior nodes of degrees up to 84 on triangles and 40 on tetrahedra. Initial guesses for solving the nonlinear systems of equations needed to derive quadrature rules are generated by forming tensor-product structures on quadrilateral/hexahedral subdomains of the simplices using the Legendre-Gauss nodes on the first half of the line reference element. In combination with a methodology for node elimination, these initial guesses lead to the development of highly efficient quadrature rules, even for very high polynomial degrees. Using existing estimates of the minimum number of quadrature points for a given degree, we show that the derived quadrature rules on triangles and tetrahedra are more than 95% and 80% efficient, respectively, for almost all degrees. The accuracy of the quadrature rules is demonstrated through numerical examples.