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

摘要

Motivated by the need for efficient and accurate simulation of the dynamicsof the polar ice sheets, we design high-order finite element discretizationsand scalable solvers for the solution of nonlinear incompressible Stokesequations. We focus on power-law, shear thinning rheologies used in modelingice dynamics and other geophysical flows. We use nonconforming hexahedralmeshes and the conforming inf-sup stable finite element velocity-pressurepairings $\mathbb{Q}_k\times \mathbb{Q}^\text{disc}_{k-2}$ or $\mathbb{Q}_k\times \mathbb{P}^\text{disc}_{k-1}$. To solve the nonlinear equations, wepropose a Newton-Krylov method with a block upper triangular preconditioner forthe linearized Stokes systems. The diagonal blocks of this preconditioner aresparse 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 ofthe high-order discretization, and the application of its inverse isapproximated using algebraic multigrid with an incomplete factorizationsmoother. This preconditioner is designed to be efficient on anisotropicmeshes, which are necessary to match the high aspect ratio domains typical forice sheets. We develop and make available extensions to two libraries---ahybrid meshing scheme for the p4est parallel AMR library, and a modifiedsmoothed aggregation scheme for PETSc---to improve their support for solvingPDEs in high aspect ratio domains. In a numerical study, we find that oursolver yields fast convergence that is independent of the element aspect ratio,the occurrence of nonconforming interfaces, and of mesh refinement, and thatdepends only weakly on the polynomial finite element order. We simulate the iceflow in a realistic description of the Antarctic ice sheet derived from fielddata, and study the parallel scalability of our solver for problems with up to383M unknowns.