A Fully Implicit Second Order Method for Viscous Free Surface Stokes Flow – Application to Glacier Simulations

Abstract. We present a fully implicit finite element method for viscous free surface Stokes problems which are common in glaciology and geodynamics. Standard fully implicit iterative solvers diverge for such problems when using large time steps. We address this by incorporating a stabilization term that vanishes upon convergence. This approach enables the use of large time steps, thereby significantly improving the efficiency of simulations. Furthermore, we leverage the fully implicit solver to develop second-order time-stepping schemes that mitigate the high errors associated with first-order methods for large time steps. The methods are tested and evaluated on model domains relevant to geodynamics and glaciology.