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

Tobin Isaac, Georg Stadler, Omar Ghattas

Research output: Contribution to journalArticle

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 pairings ℚk × ℚdisc k-2 or ℚk × ℙdisc k-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.

Original languageEnglish (US)
Pages (from-to)B804-B833
JournalSIAM Journal on Scientific Computing
Volume37
Issue number6
DOIs
StatePublished - 2015

Fingerprint

Anisotropic Mesh
High-order Finite Elements
Stokes Equations
Ice
Nonlinear Equations
Aspect Ratio
Preconditioner
Aspect ratio
Finite Element
Smoothed Aggregation
Newton-Krylov Methods
Polynomials
Geophysical Flows
Sparse Approximation
Incomplete Factorization
Algebraic multigrid
Stokes System
Element Order
Adaptive Mesh Refinement
Shear Thinning

Keywords

  • Antarctic ice sheet
  • High-order finite elements
  • Ice sheet modeling
  • Multigrid
  • Newton-Krylov method
  • Nonlinear Stokes equations
  • Preconditioning
  • Shear-thinning
  • Viscous incompressible flow

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

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title = "Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics",
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 pairings ℚk × ℚdisc k-2 or ℚk × ℙdisc k-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.",
keywords = "Antarctic ice sheet, High-order finite elements, Ice sheet modeling, Multigrid, Newton-Krylov method, Nonlinear Stokes equations, Preconditioning, Shear-thinning, Viscous incompressible flow",
author = "Tobin Isaac and Georg Stadler and Omar Ghattas",
year = "2015",
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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

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 × ℚdisc k-2 or ℚk × ℙdisc k-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 × ℚdisc k-2 or ℚk × ℙdisc k-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

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KW - Ice sheet modeling

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KW - Nonlinear Stokes equations

KW - Preconditioning

KW - Shear-thinning

KW - Viscous incompressible flow

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