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.

翻译：我们把操作者先决条件框架[R. Hiptmair,Comput. Math., with Appl. 52 (2006),pp.~699-706]扩大到Petrov-Galerkin方法,同时核算在进行数字近似时出现的不同变异形式及其先决条件的参数扰动。通过考虑原始形式及其先决条件的不同扰动参数,我们的双对数抽象设置导致稳健和控制的计划。对于Hilbert空间,我们得出对迭代求解器的详尽的线性和超线性趋同估计,例如,在以低精度或等量高度压缩近似为先决条件时,我们得出了无遗线性和超线性趋同值的迭代求解器的趋同界限。