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 A1 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 ApIF⊂A1 depending on the probability distribution.