In this paper, we discuss two problems which arise when domain decomposition is used in conjunction with co-located finite volume methods for solving the incompressible Navier-Stokes equations. The first problem occurs when one tries to compute a solution to the pressure Poisson equation (PPE) over multiple grids. Two sources of interpolation error must be addressed before the PPE can be solved. One source results from the interpolation of values from one grid to the boundary of a neighboring grid, while the other, which is unique to co-located methods, results from the interpolation of velocities from cell centers to cell faces. We have found that in order to obtain a converged pressure equation, both the interpolation schemes must be very accurate. Although our code is formally only second order in space, we use interpolation schemes which are fourth order accurate for both intergrid and center-to-face interpolations. The second problem occurs when using domain decomposition to solve problems in which the geometry includes re-entrant corners. We show that the natural way to apply domain decomposition to a corner geometry results in unstable solutions. To remedy this problem, we employ L-shaped grids in our simulations.
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