Generally robust multigrid codes, like BoxMG, employ Galerkin coarsening to form the coarse grid operators. That is, if the fine grid operator is $A$, then the coarse grid operator is $RAP$, where $P$ is the interpolation operator from the coarse grid to the fine grid, and $R$ is the restriction operator from the fine grid to the coarse grid. In BoxMG, R is chosen as the transpose of P. This choice minimizes the error in the range of interpolation. Also, in BoxMG, interpolation is operator-induced, thereby approximately preserving the continuity of the normal component of the flux across interfaces. Cell-centered discretizations on a logically structured grids are readily treated with BoxMG by associating them with the corresponding dual grid. However, this approach does not preserve the cell based structure on coarser levels. Although this structure may not be important in some applications, in the case of patch- or cell-based mesh refinement, it is very desirable. Several multigrid methods have been proposed to coarsen by cells, but these methods have been either too costly or not robust with respect to fine-scale discontinuous coefficients. For example coarsening by cells, using an operator-induced piecewise bilinear interpolation for P, the transpose of P for R, and forming the coarse grid operator as $RAP$ leads to unacceptable stencil growth. In contrast, Wesseling has investigated a method that coarsens by cells, in which R is not the transpose of P, yet under certain conditions RAP is symmetric; however, his use of a non-operator-dependent P yields a method that is not as robust as BoxMG. In this paper we explore the use of multiple dual-meshes on a logically structured two-dimensional grid in conjunction with the variational framework of BoxMG to create a robust method that coarsens by cells while maintaining a 9-point coarse-grid operator on all levels. This approach readily extends to the cases of patch- and cell-based refinement.