Improving our confidence in future projections of sea-level rise requires models that can simulate ice-sheet evolution both in the future and in the geological past. A physically accurate treatment of large changes in ice-sheet geometry requires a proper treatment of processes near the margin, like grounding line dynamics, which in turn requires a high spatial resolution in that specific region, so that small-scale topographical features are resolved. This leads to a demand for computationally efficient models, where such a high resolution can be feasibly applied in simulations of 105–107 years in duration. Here, we present and evaluate a new ice-sheet model that solves the hybrid SIA–SSA approximation of the stress balance, including a heuristic rule for the grounding-line flux. This is done on a dynamic adaptive mesh which is adapted to the modelled ice-sheet geometry during a simulation. Mesh resolution can be configured to be fine only at specified areas, such as the calving front or the grounding line, as well as specified point locations such as ice-core drill sites. This strongly reduces the number of grid points where the equations need to be solved, increasing the computational efficiency. A high resolution allows the model to resolve small geometrical features, such as outlet glaciers and sub-shelf pinning points, which can significantly affect large-scale ice-sheet dynamics. We show that the model reproduces the analytical solutions or model intercomparison benchmarks for a number of schematic ice-sheet configurations, indicating that the numerical approach is valid. Because of the unstructured triangular mesh, the number of vertices increases less rapidly with resolution than in a square-grid model, greatly reducing the required computation time for high resolutions. A simulation of all four continental ice sheets during an entire 120 kyr glacial cycle, with a 4 km resolution near the grounding line, is expected to take 100–200 wall clock hours on a 16-core system (1600–3200 core hours), implying that this model can be feasibly used for high-resolution palaeo-ice-sheet simulations.