Ed Bueler: Surface elevation errors in finite element Stokes models for glacier evolution

Abstract:

The evolution of glaciation is determined by the bedrock elevation and the surface mass balance as data. The glacier’s geometry then solves a free boundary problem. This can be posed in weak form as a variational inequality over admissible surface elevation functions, those which are above the bedrock topography. We conjecture that the fully-implicit time-step problem here is well-posed if the surface kinematical equation is appropriately regularized. This conjecture is supported by physical arguments and numerical evidence. On the other hand, we prove a general theorem which bounds the numerical error made by finite element approximations of nonlinear-operator variational inequalities in Banach spaces. The bound is a sum of error terms of different types, special to variational inequalities. In the case of implicit steps for glacier surface elevations, these terms are of three types: errors from discretizing the bed elevation, errors from numerically solving for the Stokes velocity, and finally an expected error which is quasi-optimal in the finite element representation of the surface elevation. The design of glacier evolution models is then considered, based on this error bound.