Cheap and stable quadrature on polyhedral elements

TL;DR

This paper introduces a cost-effective, stable numerical integration method for polyhedral elements that avoids tetrahedral decomposition and matrix inversion, ensuring robustness even with negative weights.

Contribution

It presents a novel tetrahedra-free quadrature approach using hyperinterpolation and Chebyshev moments, enhancing stability and reducing computational costs.

Findings

The method is stable without matrix inversion.

It avoids conditioning issues common in other approaches.

The quadrature formula remains stable with some negative weights.

Abstract

We discuss a cheap tetrahedra-free approach to the numerical integration of polynomials on polyhedral elements, based on hyperinterpolation in a bounding box and Chebyshev moment computation via the divergence theorem. No conditioning issues arise, since no matrix factorization or inversion is needed. The resulting quadrature formula is theoretically stable even in the presence of some negative weights.