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A convergence theory of multilevel additive Schwarz methods on unstructured meshes

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Abstract

We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, and the domains formed by the coarse and fine grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for nonselfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [12, 15, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the fine mesh and coarser meshes, e.g., no requirements on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the orderO(ρ2 L 2), whereρ = max1≤l≤L(h l +l− 1)/δ l,h l is the element size of thelth level mesh,δ l the overlap of subdomains on thelth level mesh, andL the number of mesh levels.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Los Angeles, 90095-1555, Los Angeles, CA, USA

    Tony F. Chan

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Jun Zou

Authors
  1. Tony F. Chan
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  2. Jun Zou
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Communicated by G. Meurant

The work was partially supported by the NSF under contract ASC 92-01266, and ONR under contract ONR-N00014-92-J-1890. The second author was also partially supported by HKRGC grants no. CUHK 316/94E and the Direct Grant of CUHK.

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Chan, T.F., Zou, J. A convergence theory of multilevel additive Schwarz methods on unstructured meshes. Numer Algor 13, 365–398 (1996). https://doi.org/10.1007/BF02207701

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  • DOI: https://doi.org/10.1007/BF02207701

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