Decomposition into subspaces preconditioning: abstract framework

  • Jakub Hrnčíř
  • Ivana PultarováEmail author
  • Zdeněk Strakoš
Original Paper


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.


Operator preconditioning Decomposition into infinite-dimensional subspaces Stable splitting Norm and spectral equivalence of operators Additive Schwarz methods Multilevel methods Separate displacement preconditioning 


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The authors thank Radim Blaheta for pointing out the separate displacement method in elasticity. The authors thank Miroslav Bulíček, Vít Dolejší, Josef Málek, Endre Süli, and Jan Zeman for their careful reading the manuscript and for many helpful suggestions and comments.

Funding information

The work was supported by the Grant Agency of the Czech Republic under the contract no. 17-04150J and by the Charles University, project GA UK no. 172915.


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Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  2. 2.Department of Mathematics, Faculty of Civil EngineeringCzech Technical University in PraguePrague 6Czech Republic

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