We discuss simulations of grain boundary motion utilizing a grain-continuum approach to representing polycrystalline films; i.e., as a system of distinct but interacting continua. Field variables are computed as functions of position in each grain; e.g., internal stresses and concentrations. Grain boundaries are represented and tracked using multiple level sets, providing a method to evolve the grain structure in time. We demonstrate the approach using stress-driven grain boundary migration in polycrystalline copper films, assuming that migration is due to vacancy migration. The anisotropic elastic constants of single crystal copper are used for each grain, under specified rotations (grain orientations). Stresses and stress gradients are thermally induced in a film with <111> texture, with the exception of a single <100> grain on a silicon dioxide film on a silicon substrate. Grain boundary velocities are calculated from the fluxes of vacancies to grain boundaries. The computed velocities are then used to update the level sets that represent the grain boundaries.