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 $O((m+1{)}^{-1})$ for elements of an approximation space ${\mathcal{A}}_1$ 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 ${\mathcal{A}}_{\infty }^{\pi }\subset {\mathcal{A}}_1$ depending on the probability distribution.

arXiv commentRevised version, accepted in Constr. Approx