CRPC-TR94460

Title:    An Algebraic Schwarz Theory

Authors:  Michael Holst

Date:     June 1994

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Keywords (list up to 8):

   Multigrid, Domain Decomposition, Schwarz,
   Convergence Theory, Operator Theory.

Abstract:

      In this paper, we discuss the general theory of additive and 
   multiplicative Schwarz methods for self-adjoint positive linear operator 
   equations, representative methods of particular interest here being 
   multigrid and domain decomposition.  We examine closely one of the most 
   useful and elegant modern convergence theories for these methods, following 
   closely the recent work of Dryja and Widlund, Xu, and their colleagues.
   Our motivation is to fully understand this theory, and then to develop a 
   variation of the theory in a slightly more general setting, which will be 
   useful in the analysis of algebraic multigrid and domain decomposition 
   methods, when little or no finite element structure is available.  Using 
   this approach we can show some convergence results for a very broad class 
   of fully algebraic domain decomposition methods, without regularity 
   assumptions about the continuous problem.  Although we cannot at this time 
   use the theory to provide a ``good'' convergence theory for algebraic 
   multigrid methods, we believe that with additional analysis it may be 
   possible to do so using this framework, as well as to use the framework to 
   guide the design of the coarse problems.  The language we employ throughout 
   is algebraic, and can be interpreted abstractly in terms of operators on 
   Hilbert spaces, or in terms of matrix operators.

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