TetraFreeQ: tetrahedra-free quadrature on polyhedral elements

TL;DR

This paper introduces TetraFreeQ, a novel algorithm for generating efficient quadrature rules on polyhedral elements without using tetrahedral subdivision, ensuring positive weights and polynomial exactness.

Contribution

The paper presents a tetrahedra-free method for constructing quadrature rules on arbitrary polyhedra, utilizing Tchakaloff discretization and Lawson-Hanson iterations, with open-source Matlab implementation.

Findings

Successfully computes low-cardinality quadrature rules

Handles arbitrary polyhedral shapes with positive weights

Validated through numerical tests

Abstract

In this paper we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape. The key tools are the notion of Tchakaloff discretization set and the solution of moment-matching equations by Lawson-Hanson iterations for NonNegative Least-Squares. Several numerical tests are presented. The method is implemented in Matlab as open-source software.