Multigrid Incomplete Factorization Methods in Krylov Subspaces on Unstructured Grids

We consider multigrid methods for solving large systems of linear algebraic equations (SLAEs) with sparse matrices arising from approximations of multidimensional boundary-value problems on unstructured grids. The approaches we propose here are based on recursive data structures for variables defined on a sequence of embedded grids, and their implementation is carried out employing approximate matrix factorization, where the forward run and the backward run correspond to, respectively, the traditional reduction stage and to the prolongation of the solution. The constructed iterative processes, depending on matrix types, are preconditioned conjugate direction methods in Krylov subspaces. Multigrid algorithms are formulated according to a recursive application of two-grid algorithms. The paper investigates the application peculiarities of the discussed approaches to the solution of two- and three-dimensional problems, including the efficiency of parallel computations on distributed and hierarchical shared memory. The efficiency of the suggested mathematical tools and software is demonstrated by the results of computational experiments on a representative series of methodological examples.