2023/08/21 by Lauritz Beck, Florian Kummer, Beck, Lauritz +1 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2308.10698
openalex publication_date 2023/08/21 · openalex created_date 2023/08/23 · openalex updated_date 2026/07/28
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.