A note on the mesh independence of convergence bounds for additive Schwarz preconditioned GMRES

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. 2002; 9 :379–397) showed with a one‐dimensional example that in the absence of a coarse grid correction the usual GMRES bound has a factor of the order of . 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 (SIAM Rev. 2005; 47 :247–272) 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 one‐dimensional example, and for several two‐dimensional model problems, are very close to each other; thus, the number of preconditioned GMRES iterations needed for convergence for a prescribed tolerance remains almost constant. Copyright © 2008 John Wiley & Sons, Ltd.