We consider the algebraic and geometric issues of the advanced parallel domain decomposition methods (DDMs) for solving very large non-symmetric systems of linear algebraic equations (SLAEs) that arise in the finite volume or the finite element approximation of the multi-dimensional boundary value problems on the non-structured grids. The main approaches in question for DDM include the balancing decomposition of the grid computational domain into parameterized overlapping or non-overlapping subdomains with different interface conditions on the internal boundaries. Also, we use two different sructures of the contacting the neigbour grid subdomains: with definition or without definition of the node dividers (separators) as the special grid subdomain. The proposed Schwarz parallel additive algorithms are based on the “total-flexible” multi-preconditioned semi-conjugate direction methods in the Krylov block subspaces. The acceleration of two-level iterative processes is provided by means of aggregation, or coarse grid correction, with different orders of basic functions, which realize a low - rank approximation of the original matrix. The auxiliary subsystems in subdomains are solved by direct or by the Krylov iterative methods. The parallel implementation of algorithms is based on hybrid programming with MPI-processes and multi-thread computing for the upper and the low levels of iterations, respectively. We describe some characteristic features of the computational technologies of DDMs that are realized within the framework of the library KRYLOV in the Institute of Computational Mathematics and Mathematical Geophysics, SB RAS, Novosibirsk. The technical requirements for this code are based on the absence of the program constraints on the degree of freedom and on the number of processor units. The conceptions of the creating the unified numerical envirement for DDMs are presented and discussed.