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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 $ω$. Using local Fourier analysis we choose $ω$ 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.