# A robust solver for viscous plastic sea ice models in a finite element framework

Subject of this talk are the mathematical challenges and the numerical treatment of large scale sea ice problems. The model under consideration goes back to Hibler ("A dynamic thermodynamic sea ice model", J. Phys. Oceanogr., Hibler 1979) and is based on a viscous-plastic description of the ice as a two-dimensional thin layer on the ocean surface.

In the first part of this talk we will derive and discuss the model in order to find a presentation that is suitable for applying modern numerical approximation tools. Then, in the second part we will focus on numerical tools for an approximation of the model with finite elements. A major numerical difficulty is the strong nonlinearity that is coming from the viscous-plastic rheology.

In a third part we present a new efficient Newton solver. The idea of the solver is to combine a fixed-point iteration (Picard solver) with a Newton method. We analytically derive the Jacobian and show its positive definiteness. The positive definiteness guarantees global convergence of a properly damped (e.g. line search) Newton iteration. The Jacobian is split into a positive definite part, which is assumed to give stable convergence and a negative semidefinite part, which might be troublesome. The negative semidefinite part is adaptively damped if convergence worsens and the Newton solver turns towards a Picard iteration. We show the improved robustness of the modified Newton solver on an idealized test case and compare it to a full Newton scheme. In every Newton step a linear system of equations must be solved. We introduce a geometric multigrid solver as preconditioner to accelerate the solution of the linear problems.