Bi-Parametric Operator Preconditioning

TL;DR

This paper extends operator preconditioning to Petrov-Galerkin methods with parameter-dependent perturbations, providing robust schemes and convergence estimates for iterative solvers in Hilbert spaces.

Contribution

It introduces a bi-parametric framework for operator preconditioning that accounts for perturbations in both forms and preconditioners, enhancing robustness and convergence analysis.

Findings

Derived linear and super-linear convergence estimates.

Established $h$-independent convergence bounds.

Validated robustness with low-accuracy preconditioners.

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.