Multigrid methods for solving two-dimensional boundary-value problems

Various methods for constructing algebraic multigrid type methods for solving multidimensional boundary-value problems are considered. Two-level iterative algorithms in Krylov subspaces based on the approximate Schur complement obtained by eliminating the edge nodes of the coarse grid are described on an example of two-dimensional rectangular grids. Some aspects of extending the methods proposed to the multilevel case, to nested triangular grids, and also to three-dimensional grids are discussed. A comparison with the classical multigrid methods based on using smoothing, restriction (aggregation), coarse-grid correction, and prolongation is provided. The efficiency of the algorithms suggested is demonstrated by numerical results for some model problems.