abstract. motivated by the need for efficient and accurate simulation of the dynamics of
the polar ice sheets, we design high-order finite element discretizations and scalable solvers for the
solution of nonlinear incompressible stokes equations. in particular, we focus on power-law, shear
thinning rheologies commonly used in modeling ice dynamics and other geophysical flows. we use
nonconforming hexahedral meshes and the conforming inf-sup stable finite element velocity-pressure
disc
pairings qk × qdisc
k−2 or qk × pk−1 , where k ≥ 2 is the polynomial order of the velocity space. to
solve the nonlinear equations, we propose a newton-krylov method with a block upper triangular
preconditioner for the linearized stokes systems. the diagonal blocks of this preconditioner are sparse
approximations of the (1,1)-block and of its schur complement. the (1,1)-block is approximated using
linear finite elements based on the nodes of the high-order discretization, and the application of its
inverse is approximated using algebraic multigrid with an incomplete factorization smoother. this
preconditioner is designed to be efficient on anisotropic meshes, which are necessary to match the
high aspect ratio domains typical for ice sheets. as part of this work, we develop and make available
extensions to two libraries—a hybrid meshing scheme for the p4est parallel adaptive mesh refinement
library, and a modified smoothed aggregation scheme for petsc—to improve their support for solving
pdes in high aspect ratio domains. in a comprehensive numerical study, we find that our solver yields
fast convergence that is independent of the element aspect ratio, the occurrence of nonconforming
interfaces, and of the mesh refinement, and that depends only weakly on the polynomial finite element
order. we simulate the ice flow in a realistic description of the antarctic ice sheet derived from field
data, and study the parallel scalability of our solver for problems with up to 383 million unknowns.
key words. viscous incompressible flow, nonlinear stokes equations, shear-thinning, high-order
finite elements, preconditioning, multigrid, newton-krylov method, ice sheet modeling, antarctic ice
sheet.