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The integration domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma \\subset \\mathbb {R}^n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0393<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>R<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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 \u201cinvariant\u201d (also known as \u201cbalanced\u201d or \u201cself-similar\u201d) measure supported on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, including in particular the Hausdorff measure <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {H}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>H<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> restricted to <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:italic>d<\/jats:italic> is the Hausdorff dimension of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are decomposed into sums of integrals over suitable partitions of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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.<\/jats:p>","DOI":"10.1007\/s11075-022-01378-9","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T12:02:43Z","timestamp":1662638563000},"page":"2071-2124","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Numerical quadrature for singular integrals on fractals"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2934-008X","authenticated-orcid":false,"given":"Andrew","family":"Gibbs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Hewett","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrea","family":"Moiola","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,8]]},"reference":[{"key":"1378_CR1","unstructured":"NIST Digital Library of Mathematical Functions. http:\/\/dlmf.nist.gov\/, release 1.1.3 of 2021-09-15"},{"key":"1378_CR2","doi-asserted-by":"publisher","first-page":"1343","DOI":"10.1109\/8.475113","volume":"43","author":"S Amari","year":"1995","unstructured":"Amari, S., Bornemann, J.: Efficient numerical computation of singular integrals with applications to electromagnetics. 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