{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/numerical-quadrature-for-singular-integrals","title":"Numerical Quadrature for Singular Integrals on Fractals","arxiv_id":"2112.11793","date":"2021-12-22","proceeding":null,"authors":["A. Gibbs","D. P. Hewett","A. Moiola"],"abstract":"We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $\\mathbb{R}^n$ is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any ``invariant'' (also known as ``balanced'' or ``self-similar'') measure supported on $\\Gamma$, including in particular the Hausdorff measure $\\mathcal{H}^d$ restricted to $\\Gamma$, where $d$ is the Hausdorff dimension of $\\Gamma$. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over $\\Gamma$ are decomposed into sums of integrals over suitable partitions of $\\Gamma$ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens.","url_abs":"https://arxiv.org/abs/2112.11793v2","url_pdf":"https://arxiv.org/pdf/2112.11793v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"numerical-quadrature-for-singular-integrals","repo_url":"https://github.com/AndrewGibbs/IFSintegrals","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}