Steady, shallow ice sheets as obstacle problems: Well-posedness and finite element approximation

Abstract

We formulate steady, shallow ice sheet flow as an obstacle problem, the unknown being the ice upper surface and the obstacle being the underlying bedrock topography. This generates a free-boundary defining the ice sheet extent. The obstacle problem is written as a variational inequality subject to the positive-ice-thickness constraint. The corresponding PDE is a highly nonlinear elliptic equation which generalizes the $p$-Laplacian equation. Our formulation also permits variable ice softness, basal sliding, and elevation-dependent surface mass balance. Existence and uniqueness are shown in restricted cases which we may reformulate as a convex minimization problem. In the general case we show existence by applying a fixed point argument. Using continuity results from that argument, we construct a numerical solution by solving a sequence of obstacle $p$-Laplacian-like problems by finite element approximation. As a real application, we compute the steady-state shape of the Greenland ice sheet in a steady present-day climate.