On the questions of parallelized Krylov's iterative methods

Mathematical questions of various computational technologies of parallelelized iterative processes of Krylov’s type for solving large sparse symmetric and non-symmetric SLAEs, obtained in grid approximations of multi-dimensional boundary value problems for PDEs, are considered. Example are presented by finite approximations in gas-hydrodynamical applications, where five unknowns in each node are defined and corresponding SLAEs have small-block structure. The base of used algorithms is flexible generalized minimal residual, FGMRES, method with dynamical preconditioners of additive type, which presents an upper level of two-step iterarive Swartz algorithm. High performance of algebraic solvers is provided by using different approaches: domain decompositions of various topologies, boundary conditions and sizes of subdomain overlapping, coarse grid correction, deflation and aggregation, and incomplete factorizations of matrices. The unified formulations of using algorithms as well as the questions of computational efficiency and scalable parallelization at the geterogenous supercomputers are described. The examples of technical requirements for peculiarities of program implementation of the libraries of parallel algorithms for solving systems of linear algebraic equation, are presented.