An algebraic multigrid method with guaranteed convergence rate

We consider the iterative solution of large sparse symmetric positive definite linear systems. We present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric M-matrices with nonnegative row sum. The coarsening is based on the aggregation of the unknowns. A key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two-grid convergence rate is bounded by a user-defined parameter. For a sensible choice of this parameter, it is shown that the recursive use of the two-grid procedure yields a convergence independent of the number of levels, provided that one uses a proper AMLI-cycle. On the other hand, the computational cost per iteration step is of optimal order if the mean aggregate size is large enough. This cannot be guaranteed in all cases but is analytically shown to hold for the model Poisson problem. For more general problems, a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method. The experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations (PDEs). The set includes two-and three-dimensional problems, with both structured and unstructured grids, some of them with local refinement and/or reentering corner, and possible jumps or anisotropies in the PDE coefficients. © 2012 Society for Industrial and Applied Mathematics.SCOPUS: ar.j