the computed approximations suffers from algebraic errors too, see [@ArioliLiiesenMiedlerStrakos13].

Having only approximate solution of , the Galerkin orthogonality of the error is violated and many standard a posteriori error estimation techniques can not be used. Therefore, the algebraic error has to be taken into account in a posteriori error analysis. Additionally, the algebraic error estimation is important for the setting of a suitable stopping criterion for iterative solvers and therefore for the optimization of the computational costs. The algebraic error estimates in the framework of energy norms were treated, e.g., in [@Arioli_NM04; @Picasso_CNME09; @VohralikStrakos10] for conforming finite element method in combination with the conjugate gradient method. In the framework of goal-oriented error estimates, a possible violation of Galerkin orthogonality was taken into account, e.g., in [@KuzminKorotov_MCS10]. The influence of the algebraic error in goal-oriented error estimates was considered in [@Meidner2009Goal] for iterative algebraic solvers (including multigrid methods). The computational error was expressed and estimated as a sum of the [discretization]{} and [algebraic]{} errors. The primal and dual algebraic problems were solved alternatively and after some number of iterations (or one multigrid cycle), the algebraic and discretization components of the errors were estimated. Then the iterative process was stopped when 