The properties of a block preconditioner that has been successfully used in finite element simulations of large scale ice-sheet flow is examined. The type of preconditioner, based on approximating the Schur complement with the mass matrix scaled by the variable viscosity, is well-known in the context of Stokes flow and has previously been analyzed for other types of non-Newtonian fluids. We adapt the theory to hold for the regularized constitutive (power-law) equation for ice and derive eigenvalue bounds of the preconditioned system for both Picard and Newton linearization using \emph{inf-sup} stable finite elements. The eigenvalue bounds show that viscosity-scaled preconditioning clusters the eigenvalues well with only a weak dependence on the regularization parameter, while the eigenvalue bounds for the traditional non-viscosity-scaled mass-matrix preconditioner are very sensitive to the same regularization parameter. The results are verified numerically in two experiments using a manufactured solution with low regularity and a simulation of glacier flow. The numerical results further show that the computed eigenvalue bounds for the viscosity-scaled preconditioner are nearly independent of the regularization parameter. Experiments are performed using both Taylor-Hood and MINI elements, which are the common choices for \emph{inf-sup} stable elements in ice-sheet models. Both elements conform well to the theoretical eigenvalue bounds, with MINI elements being more sensitive to the quality of the meshes used in glacier simulations.

翻译：本文研究了一种在大规模冰盖流动有限元模拟中成功应用的块预条件子的性质。该预条件子基于用变黏度缩放的质量矩阵近似舒尔补，在斯托克斯流中广为人知，并已针对其他类型非牛顿流体进行过分析。我们将理论推广至冰的正则化本构（幂律）方程，并推导了使用inf-sup稳定有限元进行皮卡德和牛顿线性化时预条件系统的特征值界。特征值界显示，黏度缩放预条件子能良好地聚类特征值，且对正则化参数依赖性较弱，而传统的非黏度缩放质量矩阵预条件子的特征值界对同一正则化参数非常敏感。通过两个数值实验（使用低正则性人造解和冰川流动模拟）验证了结果。数值结果进一步表明，黏度缩放预条件子的计算特征值界几乎与正则化参数无关。实验采用泰勒-胡德单元和MINI单元进行，这两种单元是冰盖模型中inf-sup稳定单元的常见选择。两种单元均与理论特征值界吻合良好，其中MINI单元对冰川模拟中使用的网格质量更为敏感。