Free-surface Stokes problem: stability estimates and time-step improvements
Accurate simulations of ice sheet dynamics, mantle convection, lava flow, and other highly viscous free-surface flows involve solving the coupled Stokes/free-surface equations. In this paper, we theoretically analyze the stability and conservation properties of the weak form of this system for Newtonian fluids and non-Newtonian fluids, at both the continuous and discrete levels. We perform the fully discrete stability analysis for finite element methods used in space with explicit and implicit Euler time-stepping methods used in time. Motivated by the theory, we propose a stabilization term designed for the explicit Euler discretization, which ensures unconditional time stability with respect to the time-step size when free-surface height is Lipschitz continuous, and permits conservation of the domain volume. Numerical experiments validate and support our theoretical findings.