New Estimates for the Rate of Convergence of the Method of Subspace Corrections

We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections. We provide upper bounds and in a special case, a lower bound for preconditioners defined via the method of successive subspace corrections. AMS subject classifications: 65F10, 65J05, 65N12, 65N55 The method of subspace corrections is a general iterative method for solving the linear system of equations arising from the variational formulation in a Hilbert space. The mod- ern theory of the subspace correction methods has showed that the multigrid method and the domain decomposition method are systematically equivalent. In this paper, we study the method of subspace corrections. We refer readers to von Neumann(7), Bramble(1), Bramble and Zhang(2), Hackbusch(4), Griebel and Oswald(3), Trottenberg, Oosterlee and Schuller(5), Xu(8,9) and Xu and Zikatanov(10) for the method of subspace correc- tions. One main focus in this paper is to provide an estimate for the rate of convergence of the method of successive subspace corrections (MSSC) in terms of the method of parallel subspace corrections (MPSC). This work can be considered as an extension and application of the convergence theory by Xu and Zikatanov(10). Based on this framework, we obtain a formula for the convergence rate, which can be employed to derive many other estimates related to the method of subspace corrections. We then show how the convergence rate of the MSSC can be estimated in terms of the MPSC. Similar results and other approaches on deriving estimates relating MSSC and MPSC are also found in earlier works(2,3,8,9).