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

• Published in: SIAM Journal on Applied Mathematics
• Main Authors: Jouvet, G., Bueler, E.
• Format: Article in Journal/Newspaper
• Language: English
• Published: SIAM 2012
• Subjects: Numerical Analysis; Greenland; Ice Sheet
• Online Access: http://publications.imp.fu-berlin.de/1867/; http://publications.imp.fu-berlin.de/1867/1/Jouvet_Bueler_2012.pdf; https://doi.org/10.1137/110856654

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.