Full-Stokes equations : solution strategies and parameter identification with application to ice sheet models

Melting of glaciers is an important factor in sea level rise. We try to estimate the future development of glaciers with simulations. They help us to prepare and evaluate the impacts of global warming on glaciers. These simulations should be as precise as possible. Thus, a high resolution and the implementation of many physical processes are necessary. The resulting large software codes need a lot of computation time and limit the resolution and the number of implemented physical processes. A realistic model should still be computationally affordable and physical parameters of the equations have to be known. Hence, we can improve predictions by developing new algorithms that reduce the computation time without reducing the accuracy of the results.

The main computation time is used to calculate the velocity and pressure field of glaciers. The governing equations are called full-Stokes equations. In this thesis, we use that minimizing a convex functional is equivalent to solving the full-Stokes equations. We prove that Newton's method with Armijo step sizes for this functional finds a sequence converging to the unique solution of the full-Stokes equations. We only have to add a small diffusion term for this theoretical analysis. We also introduce approximately exact step sizes for Newton's method and the classical Picard iteration. We show that the approximately exact step sizes are better than the Armijo step sizes for all our tested benchmark experiments.

Lastly, we focus on unknown physical parameters: The ice rheology and the friction coefficient. The known physical quantity is the surface velocity of the glacier. We show that an optimal ice rheology and an optimal friction coefficient exist. Further, we prove Gâteaux differentiability of the control-to-state operator. Finally, we prove the existence and uniqueness of a solution for the adjoint equations.

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