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

Moustapha Diallo, Zelalem Arega Worku

TL;DR

This work addresses the challenge of constructing very high-degree fully symmetric positive interior (f-SPI) quadrature rules on standard reference elements (square, cube, prism, pyramid) for high-order PDE solvers. It introduces a variable parameterization to enforce positivity and interiority, solved via Levenberg–Marquardt optimization, coupled with a symmetry-aware node-reduction strategy that combines orbit elimination and orbit collapse. The resulting rules reach degrees of $77$ on the square, $45$ on the cube, and $30$ on the prism and pyramid, often with fewer nodes than prior f-SPI rules, and come with complete node and weight data for immediate use. Numerical tests with oscillatory integrals and mesh refinement confirm accuracy and convergence comparable to existing rules, underscoring the practical impact for high-order DG and SBP discretizations.

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.

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

TL;DR

This work addresses the challenge of constructing very high-degree fully symmetric positive interior (f-SPI) quadrature rules on standard reference elements (square, cube, prism, pyramid) for high-order PDE solvers. It introduces a variable parameterization to enforce positivity and interiority, solved via Levenberg–Marquardt optimization, coupled with a symmetry-aware node-reduction strategy that combines orbit elimination and orbit collapse. The resulting rules reach degrees of on the square, on the cube, and on the prism and pyramid, often with fewer nodes than prior f-SPI rules, and come with complete node and weight data for immediate use. Numerical tests with oscillatory integrals and mesh refinement confirm accuracy and convergence comparable to existing rules, underscoring the practical impact for high-order DG and SBP discretizations.

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.
Paper Structure (15 sections, 15 equations, 17 figures, 6 tables, 2 algorithms)

This paper contains 15 sections, 15 equations, 17 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem
  3. Levenberg-Marquardt Algorithm
  4. Initial Quadrature Construction
  5. Square
  6. Cube
  7. Prism
  8. Pyramid
  9. Node Elimination
  10. Results
  11. Quadrature Rules
  12. Numerical Experiments
  13. Conclusions
  14. Nodal Distributions of Selected Quadrature Rules
  15. Supplementary Materials

Figures (17)

  • Figure 1: Visual of the domains.
  • Figure 2: Base view of the geometric construction of a degree 15 quadrature rule on the pyramid.
  • Figure 3: Geometric and algebraic initializations for degree 15 quadrature rule on the pyramid with darker nodes having larger weight values.
  • Figure 4: Efficiency on the square and cube.
  • Figure 5: Efficiency on the prism and pyramid.
  • ...and 12 more figures