Finite element approximations of a glaciology problem

Sum S. Chow; Graham F. Carey; Michael L. Anderson

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 5, page 741-756
  • ISSN: 0764-583X

Abstract

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In this paper we study a model problem describing the movement of a glacier under Glen's flow law and investigated by Colinge and Rappaz [Colinge and Rappaz, ESAIM: M2AN33 (1999) 395–406]. We establish error estimates for finite element approximation using the results of Chow [Chow, SIAM J. Numer. Analysis29 (1992) 769–780] and Liu and Barrett [Liu and Barrett, SIAM J. Numer. Analysis33 (1996) 98–106] and give an analysis of the convergence of the successive approximations used in [Colinge and Rappaz, ESAIM: M2AN33 (1999) 395–406]. Supporting numerical convergence studies are carried out and we also demonstrate the numerical performance of an a posteriori error estimator in adaptive mesh refinement computation of the problem.

How to cite

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Chow, Sum S., Carey, Graham F., and Anderson, Michael L.. "Finite element approximations of a glaciology problem." ESAIM: Mathematical Modelling and Numerical Analysis 38.5 (2010): 741-756. <http://eudml.org/doc/194237>.

@article{Chow2010,
abstract = { In this paper we study a model problem describing the movement of a glacier under Glen's flow law and investigated by Colinge and Rappaz [Colinge and Rappaz, ESAIM: M2AN33 (1999) 395–406]. We establish error estimates for finite element approximation using the results of Chow [Chow, SIAM J. Numer. Analysis29 (1992) 769–780] and Liu and Barrett [Liu and Barrett, SIAM J. Numer. Analysis33 (1996) 98–106] and give an analysis of the convergence of the successive approximations used in [Colinge and Rappaz, ESAIM: M2AN33 (1999) 395–406]. Supporting numerical convergence studies are carried out and we also demonstrate the numerical performance of an a posteriori error estimator in adaptive mesh refinement computation of the problem. },
author = {Chow, Sum S., Carey, Graham F., Anderson, Michael L.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Glen's flow law; non-Newtonian fluids; finite element error estimates; successive approximations.},
language = {eng},
month = {3},
number = {5},
pages = {741-756},
publisher = {EDP Sciences},
title = {Finite element approximations of a glaciology problem},
url = {http://eudml.org/doc/194237},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Chow, Sum S.
AU - Carey, Graham F.
AU - Anderson, Michael L.
TI - Finite element approximations of a glaciology problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 5
SP - 741
EP - 756
AB - In this paper we study a model problem describing the movement of a glacier under Glen's flow law and investigated by Colinge and Rappaz [Colinge and Rappaz, ESAIM: M2AN33 (1999) 395–406]. We establish error estimates for finite element approximation using the results of Chow [Chow, SIAM J. Numer. Analysis29 (1992) 769–780] and Liu and Barrett [Liu and Barrett, SIAM J. Numer. Analysis33 (1996) 98–106] and give an analysis of the convergence of the successive approximations used in [Colinge and Rappaz, ESAIM: M2AN33 (1999) 395–406]. Supporting numerical convergence studies are carried out and we also demonstrate the numerical performance of an a posteriori error estimator in adaptive mesh refinement computation of the problem.
LA - eng
KW - Glen's flow law; non-Newtonian fluids; finite element error estimates; successive approximations.
UR - http://eudml.org/doc/194237
ER -

References

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  1. H. Blatter, Velocity and stress fields in grounded glaciers: A simple algorithm for including deviatoric stress gradients. J. Glaciology41 (1995) 333–344.  
  2. G.F. Carey, Computational Grids: Generation, Adaptation and Solution Strategies. Taylor & Francis (1997).  Zbl0955.74001
  3. S.-S. Chow, Finite element error estimates for nonlinear elliptic equations of monotone type. Numer. Math.54 (1989) 373–393.  Zbl0643.65058
  4. S.-S. Chow, Finite element error estimates for a blast furnace gas flow problem. SIAM J. Numer. Analysis29 (1992) 769–780.  Zbl0749.76040
  5. S.-S. Chow and G.F. Carey, Numerical approximation of generalized Newtonian fluids using Heindl elements: I. Theoretical estimates. Internat. J. Numer. Methods Fluids41 (2003) 1085–1118.  Zbl1047.76040
  6. J. Colinge and H. Blatter, Stress and velocity fields in glaciers: Part I. Finite-difference schemes for higher-order glacier models. J. Glaciology44 (1998) 448–456.  
  7. J. Colinge and J. Rappaz, A strongly nonlinear problem arising in glaciology. ESAIM: M2AN33 (1999) 395–406.  Zbl0946.65115
  8. J.W. Glen, The Flow Law of Ice, Internat. Assoc. Sci. Hydrology Pub. 47, Symposium at Chamonix 1958 – Physics of the Movement of the Ice (1958) 171–183.  
  9. R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM: M2AN37 (2003) 175–186.  Zbl1046.76002
  10. W. Han, J. Soren and I. Shimansky, The Kačanov method for some nonlinear problems. Appl. Num. Anal.24 (1997) 57–79.  Zbl0878.65099
  11. C. Johnson and V. Thomee, Error estimates for a finite element approximation of a minimal surface. Math. Comp.29 (1975) 343–349.  Zbl0302.65086
  12. W.B. Liu and J.W. Barrett, Finite element approximation of some degenerate monotone quasilinear elliptic systems. SIAM J. Numer. Analysis33 (1996) 98–106.  Zbl0846.65064
  13. W.S.B. Patterson, The Physics of Glaciers, 2nd edition. Pergamon Press (1981).  
  14. E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B. Nonlinear Monotone Operators, Springer-Verlag (1990).  Zbl0684.47029

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