Exploring basal sliding with a fluidity‐based, ice‐sheet model using FOSLS
Abstract
Summary This paper develops two first‐order system—in this context, first‐order refers to the order of the PDE not to the model—least‐squares, fluidity‐based formulations of a nonlinear Stokes flow model for ice sheets that attempt to overcome the difficulties introduced by unbounded viscosity. One commonly used way to define viscosity, Glen's law, allows viscosity to become unbounded as the strain rates approach zero. Often, numerical approaches overcome these singularities by modifying viscosity to limit its maximum. The formulations in this paper, however, reframe the problem to avoid viscosity altogether by defining the system in terms of the inverse of viscosity, which is known as fluidity. This results in a quantity that approaches zero as viscosity approaches infinity. Additionally, a set of equations that represent the curl of the velocity gradient is added to help approximate the solution in a space closer to H 1 , which improves algebraic multigrid convergence. Previous research revealed that the first‐order system least‐squares formulation has difficulties in maintaining optimal discretization convergence on more complex domains. This paper discovers that this problem is linked to how the curl equations are scaled and that stronger scalings result in better solver performance but worse discretization convergence. Determining if theremore »
- Authors:
-
- Department of Applied Mathematics University of Colorado Boulder Boulder CO USA
- Department of Civil, Environmental, and Architectural Engineering University of Colorado Boulder Boulder CO USA
- Publication Date:
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1426321
- Grant/Contract Number:
- (SC) DE‐FC02‐03ER25574; (NNSA) DE‐NA0002376
- Resource Type:
- Publisher's Accepted Manuscript
- Journal Name:
- Numerical Linear Algebra with Applications
- Additional Journal Information:
- Journal Name: Numerical Linear Algebra with Applications Journal Volume: 25 Journal Issue: 3; Journal ID: ISSN 1070-5325
- Publisher:
- Wiley Blackwell (John Wiley & Sons)
- Country of Publication:
- United Kingdom
- Language:
- English
Citation Formats
Allen, Jeffery, Manteuffel, Tom, and Rajaram, Harihar. Exploring basal sliding with a fluidity‐based, ice‐sheet model using FOSLS. United Kingdom: N. p., 2018.
Web. doi:10.1002/nla.2161.
Allen, Jeffery, Manteuffel, Tom, & Rajaram, Harihar. Exploring basal sliding with a fluidity‐based, ice‐sheet model using FOSLS. United Kingdom. https://doi.org/10.1002/nla.2161
Allen, Jeffery, Manteuffel, Tom, and Rajaram, Harihar. Wed .
"Exploring basal sliding with a fluidity‐based, ice‐sheet model using FOSLS". United Kingdom. https://doi.org/10.1002/nla.2161.
@article{osti_1426321,
title = {Exploring basal sliding with a fluidity‐based, ice‐sheet model using FOSLS},
author = {Allen, Jeffery and Manteuffel, Tom and Rajaram, Harihar},
abstractNote = {Summary This paper develops two first‐order system—in this context, first‐order refers to the order of the PDE not to the model—least‐squares, fluidity‐based formulations of a nonlinear Stokes flow model for ice sheets that attempt to overcome the difficulties introduced by unbounded viscosity. One commonly used way to define viscosity, Glen's law, allows viscosity to become unbounded as the strain rates approach zero. Often, numerical approaches overcome these singularities by modifying viscosity to limit its maximum. The formulations in this paper, however, reframe the problem to avoid viscosity altogether by defining the system in terms of the inverse of viscosity, which is known as fluidity. This results in a quantity that approaches zero as viscosity approaches infinity. Additionally, a set of equations that represent the curl of the velocity gradient is added to help approximate the solution in a space closer to H 1 , which improves algebraic multigrid convergence. Previous research revealed that the first‐order system least‐squares formulation has difficulties in maintaining optimal discretization convergence on more complex domains. This paper discovers that this problem is linked to how the curl equations are scaled and that stronger scalings result in better solver performance but worse discretization convergence. Determining if there is an optimal scaling that balances performance and convergence is still an open question. Additionally, the fluidity‐based formulations are tested with three 2D benchmark problems. Two of these benchmark problems involve basal sliding and one involves a time‐dependent free surface. The fluidity‐based solutions are consistent with the standard Galerkin method using Taylor–hood elements while better resolving viscosity.},
doi = {10.1002/nla.2161},
journal = {Numerical Linear Algebra with Applications},
number = 3,
volume = 25,
place = {United Kingdom},
year = {Wed Mar 14 00:00:00 EDT 2018},
month = {Wed Mar 14 00:00:00 EDT 2018}
}
https://doi.org/10.1002/nla.2161
Web of Science
Works referenced in this record:
Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces
journal, January 2007
- Hiptmair, Ralf; Xu, Jinchao
- SIAM Journal on Numerical Analysis, Vol. 45, Issue 6
Coupled Models and Parallel Simulations for Three-Dimensional Full-Stokes Ice Sheet Modeling
journal, June 2011
- Zhang, Huai
- Numerical Mathematics: Theory, Methods and Applications, Vol. 4, Issue 3
A Multigrid Tutorial, Second Edition
book, January 2000
- Briggs, William L.; Henson, Van Emden; McCormick, Steve F.
- Other Titles in Applied Mathematics
Superplastic deformation of ice: Experimental observations
journal, June 2001
- Goldsby, D. L.; Kohlstedt, D. L.
- Journal of Geophysical Research: Solid Earth, Vol. 106, Issue B6
Well-Posedness Results for a Nonlinear Stokes Problem Arising in Glaciology
journal, January 2013
- Chen, Qingshan; Gunzburger, Max; Perego, Mauro
- SIAM Journal on Mathematical Analysis, Vol. 45, Issue 5
Results of the Marine Ice Sheet Model Intercomparison Project, MISMIP
journal, January 2012
- Pattyn, F.; Schoof, C.; Perichon, L.
- The Cryosphere, Vol. 6, Issue 3
Rate-controlling processes in the creep of polycrystalline ice
journal, October 1983
- Duval, P.; Ashby, M. F.; Anderman, I.
- The Journal of Physical Chemistry, Vol. 87, Issue 21
A parallel high-order accurate finite element nonlinear Stokes ice sheet model and benchmark experiments: A PARALLEL FEM STOKES ICE SHEET MODEL
journal, January 2012
- Leng, Wei; Ju, Lili; Gunzburger, Max
- Journal of Geophysical Research: Earth Surface, Vol. 117, Issue F1
Manufactured solutions and the verification of three-dimensional Stokes ice-sheet models
journal, January 2013
- Leng, W.; Ju, L.; Gunzburger, M.
- The Cryosphere, Vol. 7, Issue 1
Simulations of the Greenland ice sheet 100 years into the future with the full Stokes model Elmer/Ice
journal, January 2012
- Seddik, Hakime; Greve, Ralf; Zwinger, Thomas
- Journal of Glaciology, Vol. 58, Issue 209
Solution of Nonlinear Stokes Equations Discretized By High-Order Finite Elements on Nonconforming and Anisotropic Meshes, with Application to Ice Sheet Dynamics
journal, January 2015
- Isaac, Tobin; Stadler, Georg; Ghattas, Omar
- SIAM Journal on Scientific Computing, Vol. 37, Issue 6
Efficiency Based Adaptive Local Refinement for First-Order System Least-Squares Formulations
journal, January 2011
- Adler, J. H.; Manteuffel, T. A.; McCormick, S. F.
- SIAM Journal on Scientific Computing, Vol. 33, Issue 1
Achieving Textbook Multigrid Efficiency for Hydrostatic Ice Sheet Flow
journal, January 2013
- Brown, Jed; Smith, Barry; Ahmadia, Aron
- SIAM Journal on Scientific Computing, Vol. 35, Issue 2
A Matrix Dependent/Algebraic Multigrid Approach for Extruded Meshes with Applications to Ice Sheet Modeling
journal, January 2016
- Tuminaro, R.; Perego, M.; Tezaur, I.
- SIAM Journal on Scientific Computing, Vol. 38, Issue 5
Automated Solution of Differential Equations by the Finite Element Method
book, January 2012
- Logg, Anders; Mardal, Kent-Andre; Wells, Garth
- Lecture Notes in Computational Science and Engineering
Albany/FELIX : a parallel, scalable and robust, finite element, first-order Stokes approximation ice sheet solver built for advanced analysis
journal, January 2015
- Tezaur, I. K.; Perego, M.; Salinger, A. G.
- Geoscientific Model Development, Vol. 8, Issue 4
The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow
journal, July 1982
- Engelman, M. S.; Sani, R. L.; Gresho, P. M.
- International Journal for Numerical Methods in Fluids, Vol. 2, Issue 3
Benchmark experiments for higher-order and full Stokes ice sheet models (ISMIP-HOM)
journal, January 2008
- Pattyn, F.; Perichon, L.; Aschwanden, A.
- The Cryosphere Discussions, Vol. 2, Issue 1
Parallel Auxiliary Space AMG for H(Curl) Problems
journal, June 2009
- Vassilevski, Tzanio V. Kolev Panayot S.
- Journal of Computational Mathematics, Vol. 27, Issue 5
A Fluidity-Based First-Order System Least-Squares Method for Ice Sheets
journal, January 2017
- Allen, Jeffery; Leibs, Chris; Manteuffel, Tom
- SIAM Journal on Scientific Computing, Vol. 39, Issue 2
A new three-dimensional higher-order thermomechanical ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes
journal, January 2003
- Pattyn, Frank
- Journal of Geophysical Research, Vol. 108, Issue B8
Multilevel First-Order System Least Squares for Nonlinear Elliptic Partial Differential Equations
journal, January 2003
- Codd, A. L.; Manteuffel, T. A.; McCormick, S. F.
- SIAM Journal on Numerical Analysis, Vol. 41, Issue 6
Parallel finite-element implementation for higher-order ice-sheet models
journal, January 2012
- Perego, Mauro; Gunzburger, Max; Burkardt, John
- Journal of Glaciology, Vol. 58, Issue 207