A quasi-variational inequality problem arising in the modeling of growing sandpiles

John W. Barrett; Leonid Prigozhin

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 4, page 1133-1165
  • ISSN: 0764-583X

Abstract

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Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized mixed formulation involving both the primal (sand surface) and dual (sand flux) variables. We derive, analyse and compare two methods for the approximation, and numerical solution, of this mixed problem. We prove subsequence convergence of both approximations, as the mesh discretization parameters tend to zero; and hence prove existence of a solution to this mixed model and the associated regularized quasi-variational inequality problem. One of these numerical approximations, in which the flux is approximated by the divergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithm to compute not only the evolving pile surface, but also the flux of pouring sand. Results of our numerical experiments confirm the validity of the regularization employed.

How to cite

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Barrett, John W., and Prigozhin, Leonid. "A quasi-variational inequality problem arising in the modeling of growing sandpiles." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.4 (2013): 1133-1165. <http://eudml.org/doc/273227>.

@article{Barrett2013,
abstract = {Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized mixed formulation involving both the primal (sand surface) and dual (sand flux) variables. We derive, analyse and compare two methods for the approximation, and numerical solution, of this mixed problem. We prove subsequence convergence of both approximations, as the mesh discretization parameters tend to zero; and hence prove existence of a solution to this mixed model and the associated regularized quasi-variational inequality problem. One of these numerical approximations, in which the flux is approximated by the divergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithm to compute not only the evolving pile surface, but also the flux of pouring sand. Results of our numerical experiments confirm the validity of the regularization employed.},
author = {Barrett, John W., Prigozhin, Leonid},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {quasi-variational inequalities; critical-state problems; primal and mixed formulations; finite elements; existence; convergence analysis},
language = {eng},
number = {4},
pages = {1133-1165},
publisher = {EDP-Sciences},
title = {A quasi-variational inequality problem arising in the modeling of growing sandpiles},
url = {http://eudml.org/doc/273227},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Barrett, John W.
AU - Prigozhin, Leonid
TI - A quasi-variational inequality problem arising in the modeling of growing sandpiles
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 4
SP - 1133
EP - 1165
AB - Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized mixed formulation involving both the primal (sand surface) and dual (sand flux) variables. We derive, analyse and compare two methods for the approximation, and numerical solution, of this mixed problem. We prove subsequence convergence of both approximations, as the mesh discretization parameters tend to zero; and hence prove existence of a solution to this mixed model and the associated regularized quasi-variational inequality problem. One of these numerical approximations, in which the flux is approximated by the divergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithm to compute not only the evolving pile surface, but also the flux of pouring sand. Results of our numerical experiments confirm the validity of the regularization employed.
LA - eng
KW - quasi-variational inequalities; critical-state problems; primal and mixed formulations; finite elements; existence; convergence analysis
UR - http://eudml.org/doc/273227
ER -

References

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  1. [1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, Amsterdam (2003). Zbl1098.46001MR2424078
  2. [2] G. Aronson, L.C. Evans and Y. Wu, Fast/slow diffusion and growing sandpiles. J. Differ. Eqn.131 (1996) 304–335. Zbl0864.35057MR1419017
  3. [3] C. Bahriawati and C. Carstensen, Three Matlab implementations of the lowest-order Raviart–Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math.5 (2005) 333–361. Zbl1086.65107MR2194203
  4. [4] J.W. Barrett and L. Prigozhin, Dual formulations in critical state problems. Interfaces Free Bound.8 (2006) 347–368. Zbl1108.35098MR2273233
  5. [5] J.W. Barrett and L. Prigozhin, A mixed formulation of the Monge-Kantorovich equations. ESAIM: M2AN 41 (2007) 1041–1060. Zbl1132.35333MR2377106
  6. [6] J.W. Barrett and L. Prigozhin, A quasi-variational inequality problem in superconductivity. M3AS 20 (2010) 679–706. Zbl1225.35133MR2652615
  7. [7] S. Dumont and N. Igbida, On a dual formulation for the growing sandpile problem. Euro. J. Appl. Math.20 (2008) 169–185. Zbl1158.74012MR2491122
  8. [8] S. Dumont and N. Igbida, On the collapsing sandpile problem. Commun. Pure Appl. Anal.10 (2011) 625–638. Zbl1244.35068MR2754292
  9. [9] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). Zbl0322.90046MR463994
  10. [10] L.C. Evans, M. Feldman and R.F. Gariepy, Fast/slow diffusion and collapsing sandpiles. J. Differ. Eqs.137 (1997) 166–209. Zbl0879.35019MR1451539
  11. [11] M. Farhloul, A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal.18 (1998) 121–132. Zbl0909.65086MR1492051
  12. [12] G.B. Folland, Real Analysis: Modern Techniques and their Applications, 2nd Edition. Wiley-Interscience, New York (1984). Zbl0924.28001MR767633
  13. [13] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition. Springer, Berlin (1983). Zbl1042.35002MR737190
  14. [14] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York (1984). Zbl1139.65050
  15. [15] L. Prigozhin, A quasivariational inequality in the problem of filling a shape. U.S.S.R. Comput. Math. Phys.26 (1986) 74–79. MR851756
  16. [16] L. Prigozhin, A variational model of bulk solids mechanics and free-surface segregation. Chem. Eng. Sci.48 (1993) 3647–3656. 
  17. [17] L. Prigozhin, Sandpiles and river networks: extended systems with nonlocal interactions. Phys. Rev. E49 (1994) 1161–1167. MR1379784
  18. [18] L. Prigozhin, Variational model for sandpile growth. Eur. J. Appl. Math.7 (1996) 225–235. Zbl0913.73079MR1401168
  19. [19] J.F. Rodrigues and L. Santos, Quasivariational solutions for first order quasilinear equations with gradient constraint. Arch. Ration. Mech. Anal.205 (2012) 493–514. Zbl1255.49020MR2947539
  20. [20] J. Simon, Compact sets in the space Lp(0,T;B). Annal. Math. Pura. Appl.146 (1987) 65–96. Zbl0629.46031MR916688
  21. [21] J. Simon, On the existence of the pressure for solutions of the variational Navier-Stokes equations. J. Math. Fluid Mech.1 (1999) 225–234. Zbl0961.35107MR1738751
  22. [22] R. Temam, Mathematical Methods in Plasticity. Gauthier-Villars, Paris (1985). 

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