Effective numerical integration on complex shaped elements by discrete signed measures

TL;DR

This paper introduces a stable, cost-effective method for polynomial moment-based compression of multivariate measures using discrete signed measures, applicable to complex-shaped elements in numerical integration.

Contribution

It presents a novel approach that avoids matrix conditioning issues by utilizing an orthonormal basis and algebraic quadrature, with practical applications in high-order FEM/VEM and QMC integration.

Findings

Provides bounds for signed measure weights.

Demonstrates efficient quadrature on curved planar elements.

Shows compression of QMC integration on complex 3D shapes.

Abstract

We discuss a cheap and stable approach to polynomial moment-based compression of multivariate measures by discrete signed measures. The method is based on the availability of an orthonormal basis and a low-cardinality algebraic quadrature formula for an auxiliary measure in a bounding set. Differently from other approaches, no conditioning issue arises since no matrix factorization or inversion is needed. We provide bounds for the sum of the absolute values of the signed measure weights, and we make two examples: efficient quadrature on curved planar elements with spline boundary (in view of the application to high-order FEM/VEM), and compression of QMC integration on 3D elements with complex shape.