We show that a pairwise aggregation-based algebraic 2-grid method,
applied to the linear system
arising from a 1D model problem for
Poisson's equation with Dirichlet boundary conditions, reduces the
-norm of the error at each step by at least the factor
.
We then generalize this result to problems with the same eigenvectors but
different eigenvalues from the model problem, and also to problems with
different eigenvectors that are especially well-suited to the method.
Finally, we discuss the reduction in the A-norm of the error when the
2-grid method is replaced by a multigrid V-cycle and indicate that
conjugate gradient acceleration is required in order to improve the
degraded performance of multigrid V-cycle.