Decomposition into subspaces preconditioning: abstract framework

Operator preconditioning based on decomposition into subspaces has been developed in early 90’s in the works of Nepomnyaschikh, Matsokin, Oswald, Griebel, Dahmen, Kunoth, Rüde, Xu, and others, with inspiration from particular applications, e.g., to fictitious domains, additive Schwarz methods, multilevel methods etc. Our paper presents a revisited general additive splitting-based preconditioning scheme which is not connected to any particular preconditioning method. We primarily work with infinite-dimensional spaces. Motivated by the work of Faber, Manteuffel, and Parter published in 1990, we derive spectral and norm lower and upper bounds for the resulting preconditioned operator. The bounds depend on three pairs of constants which can be estimated independently in practice. We subsequently describe a nontrivial general relationship between the infinite-dimensional results and their finite-dimensional analogs valid for the Galerkin discretization. The presented abstract framework is universal and easily applicable to specific approaches, which is illustrated on several examples.

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