Adaptive reduction‐based AMG

SUMMARY With the ubiquity of large-scale computing resources has come significant attention to practical details of fast algorithms for the numerical solution of partial dierential equations. Included in this group are the class of multigrid and algebraic multigrid algorithms that are eective solvers for many of the large matrix problems arising from the discretization of elliptic operators. Algebraic multigrid (AMG) is especially eective for many problems with discontinuous coecients, discretized on unstructured grids, or over complex geometries. While much eort has been invested in improving the practical performance of AMG, little theoretical understanding of this performance has emerged. This paper presents a two-level convergence theory for a reduction-based variant of AMG, called AMGr, which is particularly appropriate for linear systems that have M-matrix-like properties. For situations where less is known about the problem matrix, an adaptive version of AMGr that automatically determines the form of the reduction needed by the AMGr process is proposed. The adaptive cycle is shown, in both theory and practice, to yield an eective AMGr algorithm. Copyright c 2006 John Wiley & Sons, Ltd.