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Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In this paper, some local multilevel methods are presented to solve the linear algebraic systems resulting from the application of adaptive conforming Bogner--Fox--Schmit rectangular element and nonconforming Adini rectangular element approximations to the biharmonic problem. The abstract Schwarz framework is applied to verify the uniform convergence of the local multilevel methods featuring Jacobi and Gauss--Seidel smoothing only on local nodes associated with the local refinements. By the abstract framework, a convergence estimate may also be derived from the stability of the space splitting and its strengthened Cauchy--Schwarz inequality. We demonstrate the optimality of the proposed algorithms by extensive numerical experiments.<\/jats:p>","DOI":"10.1137\/17m111008x","type":"journal-article","created":{"date-parts":[[2017,11,16]],"date-time":"2017-11-16T12:07:32Z","timestamp":1510834052000},"page":"A2592-A2615","source":"Crossref","is-referenced-by-count":3,"title":["Local Multilevel Methods with Rectangular Finite Elements for the Biharmonic Problem"],"prefix":"10.1137","volume":"39","author":[{"given":"Shibing","family":"Tang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuejun","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,11,16]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-003-0492-7"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/mma.1670020416"},{"key":"atypb3","volume-title":"Pitman Res. 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