  Multigrid is a powerful solver for large-scale linear systems arising from
discretized partial differential equations. The convergence theory of multigrid
methods for symmetric positive definite problems has been well developed over
the past decades, while, for nonsymmetric problems, such theory is still not
mature. As a foundation for multigrid analysis, two-grid convergence theory
plays an important role in motivating multigrid algorithms. Regarding two-grid
methods for nonsymmetric problems, most previous works focus on the spectral
radius of iteration matrix or rely on convergence measures that are typically
difficult to compute in practice. Moreover, the existing results are confined
to two-grid methods with exact solution of the coarse-grid system. In this
paper, we analyze the convergence of a two-grid method for nonsymmetric
positive definite problems (e.g., linear systems arising from the
discretizations of convection-diffusion equations). In the case of exact coarse
solver, we establish an elegant identity for characterizing two-grid
convergence factor, which is measured by a smoother-induced norm. The identity
can be conveniently used to derive a class of optimal restriction operators and
analyze how the convergence factor is influenced by restriction. More
generally, we present some convergence estimates for an inexact variant of the
two-grid method, in which both linear and nonlinear coarse solvers are
considered.
