TY - JOUR

T1 - Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics

AU - Isaac, Tobin

AU - Stadler, Georg

AU - Ghattas, Omar

N1 - Funding Information:
This work was supported by the U.S. Department of Energy Office of Science (DOE-SC), Advanced Scientific Computing Research (ASCR), Scientific Discovery through Advanced Computing (SciDAC) program, under award numbers DE-FG02-09ER25914, DE-11018096, and DE-FC02- 13ER2612, and the U.S. National Science Foundation (NSF) Cyber-enabled Discovery and Innovation (CDI) program under awards CMS-1028889 and OPP-0941678. Allocations of computing time on TACC''s Stampede under XSEDE, TG-DPP130002, and on ORNL''s Titan, which is supported by the Office of Science of the DOE under DE-AC05-00OR22725 are gratefully acknowledged. We would like to thank the three referees who reviewed this work, whose reviews greatly improved the final version. In particular, we would like to thank the two initial referees, Ray Tuminaro and one who remains anonymous. The column-preserving SA-AMG that we have developed in the DofColumns plugin was not included in the initial submission of this work but was developed in response to their insightful comments and suggestions.
Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.

PY - 2015

Y1 - 2015

N2 - 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 × ℚdisck-2 or ℚk × ℙdisck-1, where k ≥ 2 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 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.

AB - 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 × ℚdisck-2 or ℚk × ℙdisck-1, where k ≥ 2 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 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.

KW - Antarctic ice sheet

KW - High-order finite elements

KW - Ice sheet modeling

KW - Multigrid

KW - Newton-Krylov method

KW - Nonlinear Stokes equations

KW - Preconditioning

KW - Shear-thinning

KW - Viscous incompressible flow

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U2 - 10.1137/140974407

DO - 10.1137/140974407

M3 - Article

AN - SCOPUS:84953311342

VL - 37

SP - B804-B833

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 6

ER -