18th GAMM-Seminar Leipzig on
Multigrid and related methods for optimization problems

Max-Planck-Institute for Mathematics in the Sciences
Inselstr. 22-26, D-04103 [O->]Leipzig
Phone: +49.341.9959.752, Fax: +49.341.9959.999

  18th GAMM-Seminar
January, 24th-26th, 2002
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  Abstract Reinhard Nabben, Sat, 11.55-12.20 Previous Contents Next  
  An Algebraic Theory of Schwarz Methods
Reinhard Nabben (Universität Bielefeld)

Multigrid methods and domain decomposition methods are widly used for solving partial differential equations The principal advantages include enhancement of parallelism and a localized treatment. But these advantages can also be used to solve other problems. Strongly connected with domain decompostion methods and multigrid methods are the multiplicative and additive Schwarz-type methods for solving linear systems. Here we present an algebraic theory which gives a number of new convergence results. The algebraic analysis presented complements the analysis usually done on these methods using Sobolov spaces. The effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, the result of inexact local solves and of coarse grid corrections (global coarse solves) is analyzed in an algebraic setting. Moreover we give convergence results for the so-called 'restricted' Schwarz methods. These methods are widely used in practice and are the default preconditioner in the PETSc software package.

The theory is developped with Michele Benzi, Andreas Frommer and Daniel B. Szyld

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