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. We focus on power-law, shear thinning rheologies used in modeli...
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ftdatacite:10.48550/arxiv.1406.6573 2023-05-15T13:49:41+02:00 Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics Isaac, Tobin Stadler, Georg Ghattas, Omar 2014 https://dx.doi.org/10.48550/arxiv.1406.6573 https://arxiv.org/abs/1406.6573 unknown arXiv https://dx.doi.org/10.1137/140974407 arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Numerical Analysis math.NA Computational Engineering, Finance, and Science cs.CE FOS Mathematics FOS Computer and information sciences article-journal Article ScholarlyArticle Text 2014 ftdatacite https://doi.org/10.48550/arxiv.1406.6573 https://doi.org/10.1137/140974407 2022-04-01T12:52:41Z 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. We focus on power-law, shear thinning rheologies 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 $\mathbb{Q}_k\times \mathbb{Q}^\text{disc}_{k-2}$ or $\mathbb{Q}_k \times \mathbb{P}^\text{disc}_{k-1}$. To solve the nonlinear equations, 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. We develop and make available extensions to two libraries---a hybrid meshing scheme for the p4est parallel AMR library, and a modified smoothed aggregation scheme for PETSc---to improve their support for solving PDEs in high aspect ratio domains. In a 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 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 383M unknowns. : 31 pages Text Antarc* Antarctic Ice Sheet DataCite Metadata Store (German National Library of Science and Technology) Antarctic The Antarctic |
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Numerical Analysis math.NA Computational Engineering, Finance, and Science cs.CE FOS Mathematics FOS Computer and information sciences |
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Numerical Analysis math.NA Computational Engineering, Finance, and Science cs.CE FOS Mathematics FOS Computer and information sciences Isaac, Tobin Stadler, Georg Ghattas, Omar Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics |
topic_facet |
Numerical Analysis math.NA Computational Engineering, Finance, and Science cs.CE FOS Mathematics FOS Computer and information sciences |
description |
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. We focus on power-law, shear thinning rheologies 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 $\mathbb{Q}_k\times \mathbb{Q}^\text{disc}_{k-2}$ or $\mathbb{Q}_k \times \mathbb{P}^\text{disc}_{k-1}$. To solve the nonlinear equations, 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. We develop and make available extensions to two libraries---a hybrid meshing scheme for the p4est parallel AMR library, and a modified smoothed aggregation scheme for PETSc---to improve their support for solving PDEs in high aspect ratio domains. In a 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 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 383M unknowns. : 31 pages |
format |
Text |
author |
Isaac, Tobin Stadler, Georg Ghattas, Omar |
author_facet |
Isaac, Tobin Stadler, Georg Ghattas, Omar |
author_sort |
Isaac, Tobin |
title |
Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics |
title_short |
Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics |
title_full |
Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics |
title_fullStr |
Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics |
title_full_unstemmed |
Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics |
title_sort |
solution of nonlinear stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics |
publisher |
arXiv |
publishDate |
2014 |
url |
https://dx.doi.org/10.48550/arxiv.1406.6573 https://arxiv.org/abs/1406.6573 |
geographic |
Antarctic The Antarctic |
geographic_facet |
Antarctic The Antarctic |
genre |
Antarc* Antarctic Ice Sheet |
genre_facet |
Antarc* Antarctic Ice Sheet |
op_relation |
https://dx.doi.org/10.1137/140974407 |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1406.6573 https://doi.org/10.1137/140974407 |
_version_ |
1766251960231526400 |