Convergence Analysis of Perturbed Two-Grid and Multigrid Methods

We consider multigrid methods for symmetric positive definite linear systems. We present a new algebraic convergence analysis of two-grid schemes with inexact solution of the coarse grid system. This analysis allows us to bound the convergence factor of such perturbed two-grid schemes, assuming only a certain bound on the convergence factor for the unperturbed scheme (with exact solution of the coarse grid system). Applied to multigrid methods with standard W–cycle, this analysis shows that if the convergence factor of the (unperturbed) two-grid method is uniformly bounded by σ < 1/2 , then the convergence factor of the multigrid method is uniformly bounded by σ/(1 − σ) . The analysis is purely algebraic and requires only that preand post-smoothing are applied in a symmetric way. It covers both geometric and algebraic multigrid methods, and the coarse grid matrix may be of any type (not necessarily Galerkin).