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Schwarz iterative methods: Infinite space splittings

: Griebel, M.; Oswald, P.


Constructive approximation 44 (2016), Nr.1, S.121-139
ISSN: 0176-4276 (Print)
ISSN: 1432-0940 (Online)
Fraunhofer SCAI ()

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 Aπ∞⊂A1 depending on the probability distribution.