{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:04:43Z","timestamp":1776769483808,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"233","license":[{"start":{"date-parts":[[2001,6,12]],"date-time":"2001-06-12T00:00:00Z","timestamp":992304000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n In some applications of Galerkin boundary element methods one has to compute integrals which, after proper normalization, are of the form\n \n \n \n \n \n \n \u222b\n \n <\/mml:mo>\n \n a<\/mml:mi>\n <\/mml:mrow>\n \n b<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \n \u222b\n \n <\/mml:mo>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n x<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n y<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n d<\/mml:mi>\n x<\/mml:mi>\n d<\/mml:mi>\n y<\/mml:mi>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\begin{equation*}\\int _{a}^{b}\\int _{-1}^{1}\\frac {f(x,y)}{{x-y}}dxdy,\\end{equation*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n where\n \n \n \n \n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n \n \u2261\n \n <\/mml:mo>\n (<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n (a,b)\\equiv (-1,1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , or\n \n \n \n \n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n \n \u2261\n \n <\/mml:mo>\n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n (a,b)\\equiv (a,-1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , or\n \n \n \n \n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n \n \u2261\n \n <\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (a,b)\\equiv (1,b)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n \n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n f(x,y)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a smooth function. In this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a\n \n \n \n \n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n <\/mml:mrow>\n p-<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , or\n \n \n \n \n h<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n h-p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , formulation of a Galerkin method is used. This approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the Cauchy kernel, and the outer integral by a rule which takes into account the\n \n \n \n log<\/mml:mi>\n \\log<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n endpoint singularities of its integrand. Some numerical examples are also given.\n <\/p>","DOI":"10.1090\/s0025-5718-00-01272-2","type":"journal-article","created":{"date-parts":[[2005,7,11]],"date-time":"2005-07-11T17:02:26Z","timestamp":1121101346000},"page":"251-267","source":"Crossref","is-referenced-by-count":4,"title":["Error estimates in the numerical evaluation of some BEM singular integrals"],"prefix":"10.1090","volume":"70","author":[{"given":"G.","family":"Mastroianni","sequence":"first","affiliation":[]},{"given":"G.","family":"Monegato","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,6,12]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[1] A.Aimi, M.Diligenti, G.Monegato, Numerical integration schemes for the BEM solution of hypersingular integral equations, Int. J. Numer. Meth. 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