Convergence of algebraic multigrid methods for symmetric positive definite matrices with weak diagonal dominance (Q1180601)

The paper improves convergence results for the Jacobi and Gauss-Seidel relaxations, previously obtained by \textit{J. W. Ruge} and \textit{K. Stüben} [Algebraic multigrid methods (AMG), in Multigrid methods, S. F. McCormick (ed.), Frontiers in Applied Mathematics 3, 73-130 (1987; Zbl 0659.65094)] in the context of the algebraic multigrid method; the improvement consists in generalizing those results without the \(M\)-matrix hypothesis. This has been made possible by systematically applying an apparently new, although simple, formula for the quadratic form of a matrix. The author also shows that finite-element discretizations of parabolic equations usually lead to systems that satisfy the hypotheses in the paper but do not satisfy the \(M\)-matrix assumption.