Although semi-implicit semi-Lagrangian spectral atmospheric models have been very successful for decades, they are believed to face big challenges in the longer term. Foremost, the spectral method relies heavily on data-rich global communications, which may become problematic on future massively parallel machines. This paper investigates how the Helmholtz problem, as it arises in the dynamical core of a semi-implicit non-hydrostatic numerical weather prediction model with a mass-based vertical coordinate and a constant-coefficient reference state, can be solved efficiently without relying on spectral transforms, by using a multigrid-preconditioned iterative solver instead. In the particular case of a limited-area geometry, the convergence rate of this iterative solver can be determineda priori, which allows us to predict the required number of iterations. This knowledge is especially valuable for an atmospheric model that is used for operational weather forecasting, because it guarantees that the model runtime stays constant from one forecast to another. Thea prioriknowledge of the convergence rate also allows us to choose the parameters of the multigrid preconditioner optimally. Weak scalability experiments show the superior scalability of this solver with respect to a spectral solver.