Analysis of a two-level method for anisotropic diffusion equations on aligned and nonaligned grids

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Abstract

This paper is on the convergence analysis for two-grid and multigrid methods for linear systems arising from conforming linear finite element discretization of the second-order elliptic equations with anisotropic diffusion. The multigrid algorithm with a line smoother is known to behave well when the discretization grid is aligned with the anisotropic direction; however, this is not the case with a nonaligned grid. The analysis in this paper is mainly focused on two-level algorithms. For aligned grids, a lower bound is given for a pointwise smoother, and this bound shows a deterioration in the convergence rate, whereas for 'maximally' nonaligned grids (with no edges in the triangulation parallel to the direction of the anisotropy), the pointwise smoother results in a robust convergence. With a specially designed block smoother, we show that, for both aligned and nonaligned grids, the convergence is uniform with respect to the anisotropy ratio and the mesh size in the energy norm. The analysis is complemented by numerical experiments that confirm the theoretical results.

Original languageEnglish (US)
Pages (from-to)832-851
Number of pages20
JournalNumerical Linear Algebra with Applications
Volume20
Issue number5
DOIs
StatePublished - Oct 1 2013

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Applied Mathematics

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