Optimal multilevel methods for graded bisection grids

Long Chen, Ricardo H. Nochetto, Jinchao Xu

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)1-34
Number of pages34
JournalNumerische Mathematik
Volume120
Issue number1-6
DOIs
StatePublished - Jan 2012

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

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