Séminaire LJKModèles et Algorithmes Déterministes: EDPMOISE

Le Jeudi 7 Février 2013 à 11h00 en Salle 1  Tour IRMA

Séminaire de Mr Guillaume JOUVET (Institut für Mathematik)

Modelling the dynamics of marine ice sheets using an adaptive multigrid method

Résumé:

Due to warming in recent decades, number of worldwide ice sheets have been retreating substantially, and are expected to shrink even more quickly in the future. In this talk, we consider a model for the time evolution of ice sheets and ice shelves. First, the slow deformation of ice  which dominates on the grounded part  is described by the Shallow Ice Approximation (SIA). Second, the fast basal sliding  which dominates on the floating part  is described by the Shallow Shelf Approximation (SSA). At each time step, we have to solve one scalar generalized pLaplace problem with obstacle and p > 2 (SIA) [3] and one vectorial pLaplace problem with 1 < p < 2 (SSA) [5]. Both problems can be advantageously rewritten by minimising suitable, convex nonsmooth energies. By exploiting such formulations, we implement a truncated nonsmooth Newton multigrid method [1]. Our approach allows a wide choice of unstructured meshes to be used and can be easily combined to mesh adaptation techniques. In practice, one needs to rene the mesh in the neighbourhood of the grounding line, which separates the grounded and the oating parts, to capture the high gradients due to the sharp changes in the dynamical regime of ice. Most of existing renement criteria are empirical and based on the distance to the grounding line. To optimize automatically the number and the position of mesh nodes, we implement a new hierarchical error estimate for the SSA equation [2]. As an illustration, we present numerical results based on the exercises of the Marine Ice Sheet Model Intercomparison Project (MISMIP) [4].
References
[1] C. Graser and R. Kornhuber. Multigrid methods for obstacle problems. J. of Comp. Math.,
27(1):1{44, 2009.
[2] C. Graser an R. Kornhuber and U. Sack. On Hierarchical Error Estimators for Timediscretized
Phase Field Models. Proceedings of ENUMATH 2009, Springer, 397406, 2010.
[3] G. Jouvet and E. Bueler. Steady, shallow ice sheets as obstacle problems: wellposedness and
nite element approximation. SIAM J. Appl. Math. 72(4), 1292{1314, 2012.
[4] F. Pattyn and 17 others. Results of the marine ice sheet model intercomparison project, MISMIP.
The Cryosphere Discussions, 6(1):267{308, 2012.
[5] C. Schoof. A variational approach to ice stream flow. J. Fluid Mech., 556, 2006.
