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In this paper we discuss some multilevel methods for discretizations of the fourth\u2010order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner\u2013Fox\u2013Schmit rectangle, the generalization of the Bramble\u2013Pasciak\u2013Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived.<\/jats:p>","DOI":"10.1002\/nla.1680020603","type":"journal-article","created":{"date-parts":[[2005,11,1]],"date-time":"2005-11-01T19:46:29Z","timestamp":1130874389000},"page":"487-505","source":"Crossref","is-referenced-by-count":12,"title":["Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements"],"prefix":"10.1002","volume":"2","author":[{"given":"Peter","family":"Oswald","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2005,7,8]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Finite Element Solution of Boundary Value Problems","author":"Axelsson O.","year":"1984"},{"key":"e_1_2_1_3_2","volume-title":"Integral Representations of Functions and Imbedding Theorems","author":"Besov O. 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