{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:45:28Z","timestamp":1759837528285},"reference-count":10,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote by<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msub><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>the mesh width of a curved edge<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:msub><mml:mrow><mml:mi>\u0393<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u2009<\/mml:mo><mml:mo>\u2009<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>d<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>O<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">3<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for all mesh widths<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msub><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>is obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>O<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>max<\/mml:mi><mml:mo>\u2061<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">5<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.<\/jats:p>","DOI":"10.1155\/2014\/812505","type":"journal-article","created":{"date-parts":[[2014,7,10]],"date-time":"2014-07-10T21:15:06Z","timestamp":1405026906000},"page":"1-7","source":"Crossref","is-referenced-by-count":1,"title":["Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons"],"prefix":"10.1155","volume":"2014","author":[{"given":"Hu","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanying","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"98","reference":[{"key":"14","doi-asserted-by":"publisher","DOI":"10.1216\/JIE-1988-1-4-549"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/8.1.105"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.2307\/2008686"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.enganabound.2004.09.005"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01061258"},{"key":"1","year":"1971"},{"key":"8","first-page":"359","volume-title":"A new variable transformation for numerical integration","year":"1993"},{"key":"17","year":"1984"},{"issue":"1","key":"3","first-page":"9","volume":"24","year":"2006","journal-title":"Journal of Computational Mathematics"},{"key":"4","first-page":"45","volume":"1","year":"1983","journal-title":"Journal of Computational Mathematics"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/812505.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/812505.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/812505.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,29]],"date-time":"2016-07-29T09:09:28Z","timestamp":1469783368000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/812505\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":10,"alternative-id":["812505","812505"],"URL":"https:\/\/doi.org\/10.1155\/2014\/812505","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}