Multigrid Methods for Variational Problems: General Theory for the V-Cycle

In two earlier papers [SIAM J. Numer. Anal., 19 (1982), pp. 924–929; 21 (1984), pp. 255–262], we developed an algebraic convergence theory for a class of multigrid methods applied to positive definite self-adjoint linear operator equations. The purpose of the present paper is to extend these results by eliminating an earlier approximation order restriction, developing additional rate estimates and allowing for very general relaxation schemes, including those that are nonstationary and nonsymmetric. These results apply to most well-known iterative methods and preconditioners.