Multigrid methods of macro-grid domain decomposition

We consider integrated multigrid domain decomposition methods (DDM-MG) for solving large systems of linear algebraic equations (SLAEs) with sparse symmetric or asymmetric matrices and multivariate boundary value problems obtained by grid approximations. The proposed algorithms are based on the construction of single-layer or two-layer macrogrids and special ordering of nodes according to their belonging to different topological primitives of the macrogrid: macro nodes, macro edges, macro faces and subareas. At coordinated numbering of vector components, the SLAU matrix in the three-dimensional case takes a block-tri-diagonal form of the fourth order. For its solution we use some method of approximate filtering in Krylov subspaces. At the same time, the solution of auxiliary systems in subspaces is carried out by multigrid methods of block incomplete factorization, on the basis of similar topology-oriented ordering of nodes, but not at the macro-, but at the micro-level, resulting in the formation of a single preconditioner of recursive-nested type. The justification of the proposed methods is carried out for Stiltjes-type matrices.