Published

Optimal Multilevel Methods on Graded Bisection Grids

L. Chen, R. H. Nochetto, and J. Xu

Numerische Mathematik, 120(1), 1-34, 2011.

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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.