We present a simple and effective multigrid-based Poisson solver of second-order accuracy in both gravitational potential and forces in terms of the one, two, and infinity norms. The method is especially suitable for numerical simulations using nested mesh refinement. The Poisson equation is solved from coarse to fine levels using a one-way interface scheme. We introduce antisymmetrically linear interpolation for evaluating the boundary conditions across the multigrid hierarchy. The spurious forces commonly observed at the interfaces between refinement levels are effectively suppressed. We validate the method using two- and three-dimensional density-force pairs that are sufficiently smooth for probing the order of accuracy.[Full article]