Notes on convergence of an algebraic multigrid method

The convergence theory for algebraic multigrid (AMG) algorithms proposed in Chang and Huang [Q.S. Chang, Z.H. Huang, Efficient algebraic multigrid algorithms and their convergence, SIAM J. Sci. Comput. 24 (2002) 597–618] is further discussed and a smaller and elegant upper bound is obtained. On the basis of element-free AMGe [V.E. Henson, P.S. Vassilevski, Element-free AMGe: General algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput. 23(2) (2001) 629–650] we rewrite the interpolation operator for the classical AMG (cAMG), present a uniform expression and then, by introducing a sparse approximate inverse in the Frobenius norm, give a general convergence theorem which is suited for not only cAMG but also AMG for finite elements and element-free AMGe.