  Multigrid is one of the most efficient methods for solving large-scale linear
systems that arise from discretized partial differential equations. As a
foundation for multigrid analysis, two-grid theory plays an important role in
motivating and analyzing multigrid algorithms. For symmetric positive definite
problems, the convergence theory of two-grid methods with exact solution of the
Galerkin coarse-grid system is mature, and the convergence factor of exact
two-grid methods can be characterized by an identity. Compared with the exact
case, the convergence theory of inexact two-grid methods (i.e., the coarse-grid
system is solved approximately) is of more practical significance, while it is
still less developed in the literature (one reason is that the error
propagation matrix of inexact coarse-grid correction is not a projection). In
this paper, we develop a theoretical framework for the convergence analysis of
inexact two-grid methods. More specifically, we present two-sided bounds for
the energy norm of the error propagation matrix of inexact two-grid methods,
from which one can readily obtain the identity for exact two-grid convergence.
As an application, we establish a unified convergence theory for multigrid
methods, which allows the coarsest-grid system to be solved approximately.
