High-Order Symmetric Positive Interior Quadrature Rules on Two and Three Dimensional Domains

TL;DR

This paper develops high-degree symmetric positive interior quadrature rules for 2D and 3D domains, achieving fewer nodes and high accuracy, which are crucial for high-order PDE discretizations.

Contribution

It introduces a novel construction method combining parameterization, optimization, and symmetry reduction to produce high-degree f-SPI rules with fewer nodes.

Findings

Achieved degrees up to 77 on the square and 45 on the cube.

Fewer nodes than existing rules for most degrees.

Verification confirms comparable accuracy to prior rules.

Abstract

Fully symmetric positive interior (f-SPI) quadrature rules are key building blocks for high-order discretizations of partial differential equations, yet high-degree rules with few nodes remain scarce on reference elements commonly used in mesh generation. We construct new f-SPI rules on the square, cube, prism, and pyramid by coupling a variable parameterization that enforces positivity and interiority with an efficient Levenberg-Marquardt optimization and a symmetry-aware node-reduction strategy that eliminates and collapses orbits, allowing transitions between symmetry types. The resulting rules achieve degrees up to 77 on the square, 45 on the cube, and 30 on the prism and pyramid, and for most degrees use fewer nodes than previously published f-SPI quadrature rules. Verification tests demonstrate comparable accuracy to existing rules. Complete node and weight data are also provided.