  We consider the numerical solution of Poisson's equation on structured grids
using geometric multigrid with nonstandard coarse grids and coarse level
operators. We are motivated by the problem of developing high-order accurate
numerical solvers for elliptic boundary value problems on complex geometry
using overset grids. Overset grids are typically dominated by large Cartesian
background grids and thus fast solvers for Cartesian grids are highly desired.
For flexibility in grid generation we would like to consider coarsening factors
other than two, and lower-order accurate coarse-level approximations. We show
that second-order accurate coarse-level approximations are very effective for
fourth- or sixth-order accurate fine-level finite difference discretizations.
We study the use of different Galerkin and non-Galerkin coarse-level operators.
We use red-black smoothers with a relaxation parameter $\omega$. Using local
Fourier analysis we choose $\omega$ and the coarse-level operators to optimize
the overall multigrid convergence rate. Motivated by the use of red-black
smoothers in one dimension that can result in a direct solver for the standard
second-order accurate discretization to Poisson's equation, we show that this
direct-solver property can be extended to two dimensions using a rotated grid
that results from red-black coarsening. We evaluate the use of red-black
coarsening in more general settings. We also study grid coarsening by a general
factor and show that good convergence rates are retained for a range of
coarsening factors near two. We ask the question of which coarsening factor
leads to the most efficient algorithm.
