Adaptive Multigrid Methods for coupled ice sheet and ice shelf models

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For more than one decade, Antarctic ice sheets have been retreating dramatically, and are expected to shrink even more quickly in the future. Recent studies have identified the migration of the grounding line separating the grounded part of ice sheets and the floating part of ice shelves as being a key process controlling marine ice sheet stability. It is also well-known that the grounding line is very sensitive with respect to perturbations. However, existing simulation packages such as PISM are based on uniform finite difference meshes and regularization techniques for the algebraic solution. These features limit the manageable complexity of the underlying model as well as the accessible accuracy of numerical approximation. The goal of this project is to develop efficient and reliable numerical solvers for coupled polythermal ice sheet and ice shelf models that allow for higher accuracy and thus higher predictive capability of numerical simulations of Antarctic icing and deicing than existing codes. We will consider a coupled model based on the so-called shallow ice approximation (SIA) for the slow deformation of ice, and the shallow shelf equation (SSA) for the fast basal sliding. In a second step we will add a Stefan–Signorini model for the ice temperature field. Time discretization of SIA and of the Stefan–Signorini model will be based on the method of characteristics which not only provides stability and mass conservation but also gives rise to symmetric spatial problems which can be formulated in terms of convex minimization. Hence, monotone multigrid or related non-smooth Newton multigrid techniques can be applied for efficient and robust algebraic solution. This approach treats non-smoothness by convexity rather than by regularization. High resolution of the grounding line will be provided throughout the evolution by adaptive mesh refinement based on suitable a posteriori error estimates. Implementations will be carried out in the framework of the finite element environment Dune.