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How Scalable is Domain Decomposition in Practice?
"... The convergence rates and, therefore, the overall parallel efficiencies of additive Schwarz methods are often dependent on subdomain granularity. Except when ..."
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The convergence rates and, therefore, the overall parallel efficiencies of additive Schwarz methods are often dependent on subdomain granularity. Except when
ON MESH INDEPENDENCE OF CONVERGENCE BOUNDS FOR ADDITIVE SCHWARZ PRECONDITIONED GMRES ∗
"... Abstract. Additive Schwarz preconditioners, when including a coarse grid correction, are said to be optimal for certain discretized partial differential equations, in the sense that bounds on the convergence of iterative methods are independent of the mesh size h. Cai and Zou [Numer. Linear Algebra ..."
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Abstract. Additive Schwarz preconditioners, when including a coarse grid correction, are said to be optimal for certain discretized partial differential equations, in the sense that bounds on the convergence of iterative methods are independent of the mesh size h. Cai and Zou [Numer. Linear Algebra Appl., 9:379–397, 2002] showed with a onedimensional example that in the absence of a coarse grid correction the usual GMRES bound has a factor of the order of 1 / √ h. In this paper we consider the same example and show that for that example the behavior of the method is not well represented by the above mentioned bound: We use an a posteriori bound for GMRES from [Simoncini and Szyld, SIAM Rev., 47:247–272, 2005] and show that for that example a relevant factor is bounded by a constant. Furthermore, for a sequence of meshes, the convergence curves for that onedimensional example, and for several twodimensional model problems, are very close to each other, and thus the number of preconditioned GMRES iterations needed for convergence for a prescribed tolerance remains almost constant. Key words. Linear systems, additive Schwarz Preconditioning, GMRES, discretized differential equations, convergence dependence on mesh size AMS subject classifications. 65F10, 65M99, 65N22. 1. Introduction. We
minimal residual methods with Euclidean and energy norms
, 2006
"... Optimal left and right additive Schwarz preconditioning for ..."
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