Understanding the dynamics of marine ice sheets is integral to studying the evolution of the Antarctic ice sheet in both the short and long terms. An important component of the dynamics, grounding line migration, has proved difficult to represent in numerical models, and most successful attempts have made use of techniques that are only readily applicable to flow line models. However, to capture the stress transmission involved in another important component, the buttressing of a marine ice sheet by its ice shelf, the transverse direction must also be resolved. We introduce a model that solves the time-dependent shelfy stream equations and makes use of mesh adaption techniques to overcome the difficulties typically associated with the numerics of grounding line migration. We compare the model output with a recent benchmark for flow line models and show that our model yields an accurate solution while using far less resources than would be required without mesh adaption. We also show that the mesh adapting techniques extend to two horizontal dimensions. Experiments are carried out to determine how both ice shelf buttressing and ice rises affect the marine instability predicted for an ice sheet on a foredeepened bed. We find that buttressing is not always sufficient to stabilize such a sheet but collapse of the grounded portion is still greatly delayed. We also find that the effect of an ice rise is similar to that of narrowing the ice shelf.