Theoretical results on a block preconditioner used in ice-sheet modeling: eigenvalue bounds for singular power-law fluids

TL;DR

This paper analyzes a block preconditioner for ice-sheet modeling, deriving eigenvalue bounds for regularized power-law fluids, and demonstrates its effectiveness and robustness through numerical experiments.

Contribution

It adapts existing eigenvalue analysis to power-law fluids in ice-sheet models, showing viscosity-scaled preconditioning's superior eigenvalue clustering and robustness.

Findings

Viscosity-scaled preconditioning clusters eigenvalues effectively.

Eigenvalue bounds are nearly independent of regularization parameters.

Numerical experiments confirm theoretical predictions with common finite elements.

Abstract

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.