Adaptive Multigrid Methods for coupled ice sheet and ice shelf models

十多年来，南极冰盖一直在急剧退缩，预计未来还会更快地萎缩。近年来的研究发现，分隔冰架接地部分和冰架漂浮部分的接地线的迁移是控制海洋冰盖稳定性的关键过程。同样众所周知的是，接地线对扰动非常敏感。然而，现有的仿真软件包如PISM是基于均匀有限差分网格和正则化技术的代数解。这些特征限制了底层模型的可管理复杂性以及数值近似的可访问精度。该项目的目标是为耦合的多热冰盖和冰架模型开发高效可靠的数值求解器，使南极结冰和除冰的数值模拟比现有代码具有更高的精度和更高的预测能力。我们将考虑一个耦合模型，该模型基于所谓的浅冰近似（SIA），用于冰的缓慢变形，而浅陆架方程（SSA）用于快速基底滑动。在第二步中，我们将为冰的温度场添加一个Stefan-Signorini模型。SIA和Stefan-Signorini模型的时间离散化将基于特征方法，该方法不仅提供稳定性和质量守恒，而且还会产生可以用凸最小化表示的对称空间问题。因此，单调多重网格或相关的非光滑牛顿多重网格技术可用于高效、鲁棒的代数求解。这种方法通过凸性而不是正则化来处理非光滑性。基于合适的后验误差估计，自适应网格细化将在整个进化过程中提供高分辨率的接地线。实现将在有限元环境Dune的框架内进行。

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.