Iterative methods for domain decomposition methods in Krylov subspaces to solve large systems of linear algebraic equations arising from grid approximations of multidimensional boundary value problems are considered. The algorithms under study are based on purely algebraic approaches with special variants of approximate factorization of matrices arizing from grid division by a single-layer or two-layer separating macrogrid subsets. Traditional interface boundary conditions between contacting subdomains are replaced by matrix approximations with the compensation principle exploiting. The implementation of preconditioning matrices is carried out by naturally parallelizable forward and back sweep algorithms on the macrogrid. The issues of assessing the efficiency and performance of the proposed methods and technologies for two-dimensional and three-dimensional problems are discussed, including the cases of parallelizing calculations. The results of numerical experiments for a set of methodical problems are presented.