Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics

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 pairings Qk × Qdisc k−2 or Qk × P disc k−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.