Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids

Tony Chan*, Barry F. Smith, Jun Zou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

49 Scopus citations

Abstract

We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption on the substructures that constitute the whole domain, so the substructures can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved, and neither the coarse mesh nor the fine mesh need be quasi-uniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate as the usual two level overlapping domain decomposition methods on structured meshes. The condition number of the preconditioned system depends only on the (possibly small) overlap of the substructures and the size of the coarse grid, but is independent of the sizes of the subdomains.

Original languageEnglish (US)
Pages (from-to)149-167
Number of pages19
JournalNumerische Mathematik
Volume73
Issue number2
DOIs
StatePublished - Jan 1 1996

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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