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\iteman{io-port 06111120}
\itemau{Jouvet, Guillaume; Bueler, Ed}
\itemti{Steady, shallow ice sheets as obstacle problems: well-posedness and finite element approximation.}
\itemso{SIAM J. Appl. Math. 72, No. 4, 1292-1314 (2012).}
\itemab
Summary: 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.
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\itemut{ice sheet model; shallow ice approximation; obstacle problem; variational inequality; $p$-Laplace; finite elements}
\itemli{doi:10.1137/110856654}
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