Bi-Parametric Operator Preconditioning
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We extend the general operator preconditioning framework [R. Hiptmair, Comput. Math. with Appl. 52 (2006), pp. 699-706] to account for parameter-dependent perturbations of variational forms and their preconditioning. These perturbations can arise, for example, from quadrature approximation or machine precision, when solving variational formulations as in standard finite or boundary element methods. By explicitly considering different perturbation parameters for the original equation and its preconditioning, our bi-parametric abstract setting leads to endomorphisms. Furthermore, thanks to this novel approach, new techniques to enhance iterative linear solvers performance are deduced. Further implications of our results for elliptic operators and second-kind Fredholm operators are given.
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