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.