The motion of glaciers can be simulated with the p-Stokes equations. We present an algorithm that solves these equations faster than the Picard iteration. We do that by proving q-superlinear 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 also consider approximations of exact step sizes. Exact step sizes are possible because we reformulate the problem as minimizing a convex functional. Next, we prove that the additional diffusion term only causes minor differences in the solution compared to the original p-Stokes equations. Finally, we test our algorithms on a reformulation of the experiment ISMIP-HOM B. The approximation of exact step sizes for the Picard iteration and Newton's method is superior in the experiment compared to the Picard iteration. Also, Newton's method with Armijo step sizes converges faster than the Picard iteration. However, the reached accuracy of Newton's method with Armijo step sizes depends more on the resolution of the domain.

翻译：冰川运动可通过p-Stokes方程进行模拟。本文提出一种比Picard迭代更快速求解该方程的算法。我们通过证明带Armijo步长的无穷维牛顿法对该方程解具有全局q-超线性收敛性来实现这一目标。为获得此收敛结果，我们仅需添加任意小的扩散项。同时考虑了精确步长的近似方案。由于将问题重新表述为凸泛函极小化问题，精确步长成为可能。进一步证明，相较于原始p-Stokes方程，额外扩散项仅对解造成微小差异。最后，我们在重新表述的ISMIP-HOM B实验上测试算法。实验表明，Picard迭代与牛顿法的精确步长近似策略均优于传统Picard迭代。同时，带Armijo步长的牛顿法收敛速度优于Picard迭代，但其所能达到的精度更依赖于计算域的分辨率。