Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology

Roland Glowinski; Jacques Rappaz

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 1, page 175-186
  • ISSN: 0764-583X

Abstract

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The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear elliptic problems arising in glaciology. The stationary motion of a glacier is given by a non-Newtonian fluid flow model which becomes, in a first two-dimensional approximation, the so-called infinite parallel sided slab model. The approximation of this model is made by a finite element method with piecewise polynomial functions of degree 1. Numerical results show that the theoretical results we have obtained are almost optimal.

How to cite

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Glowinski, Roland, and Rappaz, Jacques. "Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 175-186. <http://eudml.org/doc/194152>.

@article{Glowinski2010,
abstract = { The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear elliptic problems arising in glaciology. The stationary motion of a glacier is given by a non-Newtonian fluid flow model which becomes, in a first two-dimensional approximation, the so-called infinite parallel sided slab model. The approximation of this model is made by a finite element method with piecewise polynomial functions of degree 1. Numerical results show that the theoretical results we have obtained are almost optimal. },
author = {Glowinski, Roland, Rappaz, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite element method; a priori error estimates; a posteriori error estimates; non-Newtonian fluids; infinite parallel sided slab model in glaciology.; finite element method; a priori error estimates; a posteriori error estimates},
language = {eng},
month = {3},
number = {1},
pages = {175-186},
publisher = {EDP Sciences},
title = {Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology},
url = {http://eudml.org/doc/194152},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Glowinski, Roland
AU - Rappaz, Jacques
TI - Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 1
SP - 175
EP - 186
AB - The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear elliptic problems arising in glaciology. The stationary motion of a glacier is given by a non-Newtonian fluid flow model which becomes, in a first two-dimensional approximation, the so-called infinite parallel sided slab model. The approximation of this model is made by a finite element method with piecewise polynomial functions of degree 1. Numerical results show that the theoretical results we have obtained are almost optimal.
LA - eng
KW - Finite element method; a priori error estimates; a posteriori error estimates; non-Newtonian fluids; infinite parallel sided slab model in glaciology.; finite element method; a priori error estimates; a posteriori error estimates
UR - http://eudml.org/doc/194152
ER -

References

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  1. J. Baranger and H. El Amri. Estimateurs a posteriori d'erreurs pour le calcul adaptatif d'écoulements quasi-newtoniens. RAIRO Modél. Math. Anal. Numér. 25 (1991) 31-48.  
  2. J.W. Barrett and W. Liu, Finite element approximation of degenerate quasi-linear elliptic and parabolic problems. Pitman Res. Notes Math. Ser.303 (1994) 1-16. In Numerical Analysis 1993.  
  3. H. Blatter, Velocity and stress fields in grounded glacier: a simple algorithm for including deviator stress gradients. J. Glaciol.41 (1995) 333-344.  
  4. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Stud. Math. Appl. 4 (1978).  
  5. J. Colinge and J. Rappaz, A strongly non linear problem arising in glaciology. ESAIM: M2AN33 (1999) 395-406.  
  6. R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un, et la résolution par pénalisation-dualité, d'une classe de problèmes de Dirichlet non linéaires. Anal. Numér.2 (1975) 41-76.  
  7. P. Hild, I.R. Ionescu, T. Lachand-Robert and I. Rosca, The blocking of an inhomogeneous Bingham fluid. Applications to landslides. ESAIM: M2AN36 (2002) 1013-1026.  
  8. W. Liu and N. Yan. Quasi-norm local error estimators for p-Laplacian. SIAM J. Numer. Anal. 39 (2001) 100-127.  
  9. A. Reist, Résolution numérique d'un problème à frontière libre issu de la glaciologie. Diploma thesis, Department of Mathematics, EPFL, Lausanne, Switzerland (2001).  

Citations in EuDML Documents

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  1. Sum S. Chow, Graham F. Carey, Michael L. Anderson, Finite element approximations of a glaciology problem
  2. Silvia Jimenez, Correctors and field fluctuations for the pϵ(x)-laplacian with rough exponents : The sublinear growth case
  3. Marco Picasso, Jacques Rappaz, Adrian Reist, Numerical simulation of the motion of a three-dimensional glacier
  4. Jérôme Droniou, Robert Eymard, Thierry Gallouët, Raphaèle Herbin, A unified analysis of elliptic problems with various boundary conditions and their approximation
  5. Sum S. Chow, Graham F. Carey, Michael L. Anderson, Finite element approximations of a glaciology problem
  6. Boris Andreianov, Franck Boyer, Florence Hubert, Finite volume schemes for the p-laplacian on cartesian meshes
  7. Boris Andreianov, Franck Boyer, Florence Hubert, Finite volume schemes for the p-Laplacian on Cartesian meshes
  8. Jérôme Droniou, Finite volume schemes for fully non-linear elliptic equations in divergence form
  9. Jérôme Droniou, Finite volume schemes for fully non-linear elliptic equations in divergence form

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