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We found characteristics that can help us predict the convergence or divergence of iterative aggregationdisaggregation methods. We provided two results for spectral radii of asymptotic error propagation matrices: (i) the spectral radius is bounded by unity for symmetric Markov chains and (ii) the spectral radius can be arbitrarily large for a certain class of sparse Markov chains. Surprisingly, permuting states of cyclic Markov chains by algorithms usually used for reducing bandwidth of matrices leads to the latter case. We proposed a sorting method that prevents divergence for this class of Markov chains.

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We propose a specialized preconditioned FE homogenization scheme that is significantly more efficient than generic FE implementations.• Our formulation offers computational efficiency equivalent to spectral homogenization solvers.• Our formulation is devoid of ringing phenomena intrinsic to spectral methods.• All results are reproducible in an open-source project µSpectre available at https: //muspectre.gitlab.io/muspectre.