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The completion of locally refined simplicial partitions created by bisection

Author: Rob Stevenson
Journal: Math. Comp. 77 (2008), 227-241
MSC (2000): Primary 65N50, 65Y20, 65N30
Published electronically: July 26, 2007
MathSciNet review: 2353951
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Abstract: Recently, in [Found. Comput. Math., 7(2) (2007), 245-269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466-488] by Morin, Nochetto, and Siebert, converges with the optimal rate.The number of triangles $ N$ in the output partition of such a method is generally larger than the number $ M$ of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes.A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219-268] by Binev, Dahmen and DeVore saying that $ N-N_0 \leq C M$ for some absolute constant $ C$, where $ N_0$ is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of $ n$-simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.

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Additional Information

Rob Stevenson
Affiliation: Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands
Address at time of publication: Korteweg de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Keywords: Adaptive finite element methods, conforming partitions, bisection, $n$-simplices
Received by editor(s): September 23, 2005
Received by editor(s) in revised form: May 3, 2006
Published electronically: July 26, 2007
Additional Notes: This work was supported by the Netherlands Organization for Scientific Research and by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00286.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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