  Due to the wide separation of time scales in geophysical fluid dynamics,
semi-implicit time integrators are commonly used in operational atmospheric
forecast models. They guarantee the stable treatment of fast (acoustic and
gravity) waves, while not suffering from severe restrictions on the timestep
size. To propagate the state of the atmosphere forward in time, a non-linear
equation for the prognostic variables has to be solved at every timestep. Since
the nonlinearity is typically weak, this is done with a small number of Newton-
or Picard- iterations, which in turn require the efficient solution of a large
system on linear equations with $\mathcal{O}(10^6-10^9)$ unknowns. This linear
solve is often the computationally most costly part of the model. In this paper
an efficient linear solver for the LFRic next-generation model, currently
developed by the Met Office, is described. The model uses an advanced mimetic
finite element discretisation which makes the construction of efficient solvers
challenging compared to models using standard finite-difference and
finite-volume methods. The linear solver hinges on a bespoke multigrid
preconditioner of the Schur-complement system for the pressure correction. By
comparing to Krylov-subspace methods, the superior performance and robustness
of the multigrid algorithm is demonstrated for standard test cases and
realistic model setups. In production mode, the model will have to run in
parallel on 100,000s of processing elements. As confirmed by numerical
experiments, one particular advantage of the multigrid solver is its excellent
parallel scalability due to avoiding expensive global reduction operations.
