The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting from semi-discretization of the heat equation in one and two dimensions. This theory and some heuristic arguments predict that preconditioning is most effective for the first few iterations.
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The splitting problem is studied for short exact sequences consisting of countable projective limits of DFN-spaces (*) 0 → F → X → G → 0, where F or G are isomorphic to the space of distributions D'. It is proved that every sequence (*) splits for F ≃ D' iff G is a subspace of D' and that, for ultrabornological F, every sequence (*) splits for G ≃ D' iff F is a quotient of D'
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