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Optimal multilevel methods for graded bisection grids

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Abstract

We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices - the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Long Chen

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  3. The School of Mathematical Science, Peking University, Beijing, China

    Jinchao Xu

  4. Department of Mathematics, Pennsylvania State University, University Park, PA, 16801, USA

    Jinchao Xu

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  1. Long Chen
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  2. Ricardo H. Nochetto
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Correspondence to Long Chen.

Additional information

L. Chen was supported in part by NSF Grant DMS-0505454, DMS-0811272, and in part by 2010–2011 UC Irvine Academic Senate Council on Research, Computing and Libraries (CORCL). R. H. Nochetto was supported in part by NSF Grant DMS-0505454 and DMS-0807811. J. Xu was supported in part by NSF DMS-0609727, DMS 0915153, NSFC-10528102 and Alexander von Humboldt Research Award for Senior US Scientists.

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Chen, L., Nochetto, R.H. & Xu, J. Optimal multilevel methods for graded bisection grids. Numer. Math. 120, 1–34 (2012). https://doi.org/10.1007/s00211-011-0401-4

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