High-Order Numerical Integration on Domains Bounded by Intersecting Level Sets

TL;DR

This paper introduces a high-order numerical integration method for complex domains defined by intersecting level sets, enabling accurate and efficient computation on curved geometries using mapped hypercube quadrature rules.

Contribution

The paper presents a novel high-order approach for numerical integration on implicitly defined domains using mappings from hypercubes, applicable to complex geometries without fallback methods.

Findings

High-order convergence demonstrated with smooth integrands.

Effective integration on polynomial, rational, and trigonometric level set domains.

Method compatible with adaptive quadrature techniques.

Abstract

We present a high-order method that provides numerical integration on volumes, surfaces, and lines defined implicitly by two smooth intersecting level sets. To approximate the integrals, the method maps quadrature rules defined on hypercubes to the curved domains of the integrals. This enables the numerical integration of a wide range of integrands since integration on hypercubes is a well known problem. The mappings are constructed by treating the isocontours of the level sets as graphs of height functions. Numerical experiments with smooth integrands indicate a high-order of convergence for transformed Gauss quadrature rules on domains defined by polynomial, rational, and trigonometric level sets. We show that the approach we have used can be combined readily with adaptive quadrature methods. Moreover, we apply the approach to numerically integrate on difficult geometries without requiring a low-order fallback method.