Numerical approximation of viscous contact problems applied to glacial sliding

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach. These problems are of particular importance in glaciology, where they arise in the study of grounding lines and subglacial cavities. In this work, we propose a novel numerical method for solving viscous contact problems based on a mixed formulation with Lagrange multipliers of a variational inequality involving the Stokes equation. The advection equation for evolving the geometry of the domain occupied by the fluid is then solved via a specially-built upwinding scheme, leading to a robust and accurate algorithm for viscous contact problems. We then use this numerical scheme to reconstruct friction laws for glacial sliding with cavitation. We also study the effects of unsteady water pressures on sliding with cavitation.