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

TL;DR

This work develops highly accurate, fully-symmetric positive-interior quadrature rules on triangles and tetrahedra, achieving degrees up to $84$ and $40$ respectively by combining Line-LG initial guesses with a node-elimination workflow. The approach converts LG nodes from the line reference into simplex subdomains, solves a nonlinear least-squares system with Levenberg–Marquardt to produce line-LG rules, and then refines them to new, more efficient rules while preserving interior positivity. Key contributions include extending prior high-order PI rules, quantifying efficiency against established lower-bound estimates, and demonstrating practical accuracy for highly oscillatory integrals with convergence rates near $p+2$ for even and $p+1$ for odd degrees. The results enhance high-order discretizations and SBP-operator construction in PDE solvers, with publicly available supplementary data for broader use.

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.

Figures (6)

• Figure 1: Algorithm flowchart for generating symmetric PI quadrature rules.
• Figure 2: Generation of initial nodes for the degree $q=4$ line-LG quadrature rule on triangles and tetrahedra.
• Figure 3: Efficiency of the line-LG quadrature rules relative to existing symmetric quadrature rules on triangles (a) and tetrahedra (b).
• Figure 4: Examples of symmetric PI quadrature rules on triangles and tetrahedra.
• Figure 5: Efficiency of the quadrature rules derived in this work relative to existing symmetric quadrature rules on triangles (a) and tetrahedra (b).