Ice sheets and glaciers constitute an enormous water storage, currently corresponding to a potential sea level rise of almost 70 meters if all ice was to melt completely. The ice sheets are dynamic components of the global climate system and numerical modeling is a useful tool that can help us understand and predict how the ice sheets develop. The most accurate model available for ice sheets is given by the Stokes equations, but to solve them for a real ice sheet on a relevant time scale would be way too computationally costly. Instead approximations of the Stokes equations are used such as the Shallow Ice Approximation (SIA). The SIA is valid for areas where the aspect ratio, the ice thickness divided by the horizontal extent of the ice, is small. In this project equations for temperature and surface evolution were implemented in a Matlab version of SIA. The model already had algorithms implemented for computation of stresses, velocities and pressures for an ice sheet with fixed geometry and temperature. Implementation of temperature and free surface equations also made the problem time-dependent. The result was evaluated by solving a simple test problem and comparing the solution to a full Stokes solution obtained with the code ElmerIce. The SIA solution was closer to the Stokes solution when the aspect ratio ε and slope α were decreased simultaneously such that ε=arctanα, but a similar improvement was also obtained when only the slope was decreased. The differences between the two solutions were satisfyingly small for both temperature, surface location and velocities for an aspect ratio of ε= 7.8 10−4 and ε=arctanα.

Ice sheets, such as the Greenland Ice Sheet or Antarctic Ice Sheet, have a fundamental impact on landscape formation, the global climate system, and on sea level rise. The slow, creeping flow of ice can be represented by a non-linear version of the Stokes equations, which treat ice as a non-Newtonian, viscous fluid. Large spatial domains combined with long time spans and complexities such as a non-linear rheology, make ice sheet simulations computationally challenging. The topic of this thesis is the efficiency and error control of large simulations, both in the sense of mathematical modelling and numerical algorithms. In the first part of the thesis, approximative models based on perturbation expansions are studied. Due to a thick boundary layer near the ice surface, some classical assumptions are inaccurate and the higher order model called the Second Order Shallow Ice Approximation (SOSIA) yields large errors. In the second part of the thesis, the Ice Sheet Coupled Approximation Level (ISCAL) method is developed and implemented into the finite element ice sheet model Elmer/Ice. The ISCAL method combines the Shallow Ice Approximation (SIA) and Shelfy Stream Approximation (SSA) with the full Stokes model, such that the Stokes equations are only solved in areas where both the SIA and SSA is inaccurate. Where and when the SIA and SSA is applicable is decided automatically and dynamically based on estimates of the modeling error. The ISCAL method provides a significant speed-up compared to the Stokes model. The third contribution of this thesis is the introduction of Radial Basis Function (RBF) methods in glaciology. Advantages of RBF methods in comparison to finite element methods or finite difference methods are demonstrated. / eSSENCE