Convergence of algebraic multigrid based on smoothed aggregation

We prove a convergence estimate for the Algebraic Multigrid Method with prolongations deened by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the aggregates, which can be a-priori veriied computationally. Construction of the prolongator in the case of a general second order system is described, and the assumptions of the theorem are veriied for a scalar problem discretized by linear conforming nite elements.