Global convergence of Newton's method for the regularized $p$-Stokes equations

TL;DR

This paper proves the global convergence of Newton's method for the infinite-dimensional regularized p-Stokes equations, with applications to glacier flow simulations, and compares algorithmic performance in numerical experiments.

Contribution

It extends the analysis of Newton's method to infinite-dimensional p-Stokes equations, introducing a small diffusion term to ensure convergence and testing the method on glacier flow models.

Findings

Newton's method converges globally with Armijo step sizes when a small diffusion term is added.

Exact step sizes outperform Armijo step sizes in some experiments.

Newton's method with approximate step sizes converges faster in sliding glacier simulations.

Abstract

The motion of glaciers can be simulated with the $p$-Stokes equations. Up to now, Newton's method to solve these equations has been analyzed in finite-dimensional settings only. We analyze the problem in infinite dimensions to gain a new viewpoint. We do that by proving global convergence of the infinite-dimensional Newton's method with Armijo step sizes to the solution of these equations. We only have to add an arbitrarily small diffusion term for this convergence result. We prove that the additional diffusion term only causes minor differences in the solution compared to the original $p$-Stokes equations under the assumption of some regularity. Finally, we test our algorithms on two experiments: A reformulation of the experiment ISMIP-HOM $B$ without sliding and a block with sliding. For the former, the approximation of exact step sizes for the Picard iteration and exact step sizes and Armijo step sizes for Newton's method are superior in the experiment compared to the Picard iteration. For the latter experiment, Newton's method with Armijo step sizes needs many iterations until it converges fast to the solution. Thus, Newton's method with approximately exact step sizes is better than Armijo step sizes in this experiment.