Bi-Parametric Operator Preconditioning

2020/11/10 by Paul Escapil-Inchauspé, Escapil-Inchauspé, Paul, Carlos Jerez-Hanckes +1
Computer Science · Mathematics · #65F08 #65F10 #65N22 #65N30 #FOS: Mathematics #Numerical Analysis (math.NA) #cs.NA #math.NA #msc:65F08 #msc:65F10 #msc:65N22 #msc:65N30

paper · pdf · doi:10.48550/arxiv.2011.05028

arxiv created 2022/03/29 · arxiv updated 2022/03/30

Abstract

We extend the operator preconditioning framework [R. Hiptmair, Comput. Math. with Appl. 52 (2006), pp.~699--706] to Petrov-Galerkin methods while accounting for parameter-dependent perturbations of both variational forms and their preconditioners, as occurs when performing numerical approximations. By considering different perturbation parameters for the original form and its preconditioner, our bi-parametric abstract setting leads to robust and controlled schemes. For Hilbert spaces, we derive exhaustive linear and super-linear convergence estimates for iterative solvers, such as h-independent convergence bounds, when preconditioning with low-accuracy or, equivalently, with highly compressed approximations.

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