{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:50:37Z","timestamp":1776844237927,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"222","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We study a multilevel additive Schwarz method for the\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns\n \n \n \n M<\/mml:mi>\n M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We prove that the condition number\n \n \n \n \n \n \u03ba\n \n <\/mml:mi>\n (<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\kappa (P)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the multilevel additive Schwarz operator behaves like\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n M<\/mml:mi>\n <\/mml:msqrt>\n \n log<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\sqrt {M}\\log ^2M)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . As a direct consequence of this we also give the results for the\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -level preconditioner and also for the\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n version with quasi-uniform meshes. Numerical results supporting our theory are presented.\n <\/p>","DOI":"10.1090\/s0025-5718-98-00926-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"501-518","source":"Crossref","is-referenced-by-count":21,"title":["Multilevel additive Schwarz method for the \u210e-\ud835\udc5d version of the Galerkin boundary element method"],"prefix":"10.1090","volume":"67","author":[{"given":"Norbert","family":"Heuer","sequence":"first","affiliation":[]},{"given":"Ernst","family":"Stephan","sequence":"additional","affiliation":[]},{"given":"Thanh","family":"Tran","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"M. Ainsworth, A preconditioner based on domain decomposition for \u210e-\ud835\udc5d finite element approximation on quasi-uniform meshes, SIAM J. Numer. 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