Analysis of an Algebraic Petrov-Galerkin Smoothed Aggregation Multigrid Method

We give a convergence estimate for a Petrov-Galerkin Algebraic Multigrid method. In this method, the prolongations are defined using the concept of smoothed aggre-gation while the restrictions are simple aggregation operators. The analysis is carried out by showing that these methods can be interpreted as variational Ritz-Galerkin ones using modified transfer and smoothing operators. The estimate depends only on a weak approximation property for the aggregation operators. For a scalar sec-ond order elliptic problem using linear elements, this assumption is shown to hold using simple geometrical arguments on the aggregates