Antarctic ice sheets have been retreating dramatically in recent years, and are expected to shrink even more quickly in the future, with a subsequent rise of sea level. It is then crucial to develop accurate numerical simulation tools of ice sheets that have a high predictive capability. Modelling the dynamics of a shallow marine ice sheet requires solving a non-linear elliptic problem. The solution of such problem usually shows high gradients at the transition zones between the floating and the grounded part of the ice sheet. This feature makes the numerical approximation challenging, since such zones are very narrow compared to the entire ice sheet. Performing a simulation of a real ice sheet at a resolution which is physically admissible does not only require an efficient solver, but also a suitable adaptive mesh technique, to reduce the computational costs to a reasonable level. Despite the efforts that were made in recent years, the distribution of the mesh nodes is still empirical in most of existing models, and then far from optimal. The goal of this project is to develop an adaptive multigrid method for a shallow marine ice sheet model which combines the efficiency of monotone multigrid solvers and the adaptive mesh techniques based on hierarchical error estimates. Real-life simulations will be carried out in cooperation with glaciologists from the Potsdam Institute of Climate Impact Research.