Schwarz Iterative Methods: Infinite Space Splittings

We prove the convergence of greedy and randomized versions of Schwarz iterative methods for solving linear elliptic variational problems based on infinite space splittings of a Hilbert space. For the greedy case, we show a squared error decay rate of (O((m+1)^{-1})) for elements of an approximation space (mathscr {A}_1) related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of (O((m+1)^{-1})) on a class (mathscr {A}_{infty }^{pi }subset mathscr {A}_1) depending on the probability distribution.