Schwarz iterative methods: Infinite space splittings

Abstract

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 <em>O</em>((<em>m</em> + 1)^-1) for elements of an approximation space &#119964;₁ related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of <em>O</em>((<em>m</em> + 1)^-1) on a class &#119964;^π_∞ ⊂ &#119964;₁ depending on the probability distribution.