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

arXiv:2601.14488v1 Announce Type: new
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.