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.
Mots clés : quasi-variational inequalities, critical-state problems, primal and mixed formulations, finite elements, existence, convergence analysis
@article{M2AN_2013__47_4_1133_0,
author = {Barrett, John W. and Prigozhin, Leonid},
title = {A quasi-variational inequality problem arising in the modeling of growing sandpiles},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1133--1165},
publisher = {EDP-Sciences},
volume = {47},
number = {4},
year = {2013},
doi = {10.1051/m2an/2012062},
mrnumber = {3082292},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012062/}
}
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 PY - 2013 SP - 1133 EP - 1165 VL - 47 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012062/ DO - 10.1051/m2an/2012062 LA - en ID - M2AN_2013__47_4_1133_0 ER -
%0 Journal Article %A Barrett, John W. %A Prigozhin, Leonid %T A quasi-variational inequality problem arising in the modeling of growing sandpiles %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1133-1165 %V 47 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012062/ %R 10.1051/m2an/2012062 %G en %F M2AN_2013__47_4_1133_0
Barrett, John W.; Prigozhin, Leonid. A quasi-variational inequality problem arising in the modeling of growing sandpiles. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 4, pp. 1133-1165. doi : 10.1051/m2an/2012062. http://archive.numdam.org/articles/10.1051/m2an/2012062/
[1] and , Sobolev Spaces. Academic Press, Amsterdam (2003). | MR | Zbl
[2] , and , Fast/slow diffusion and growing sandpiles. J. Differ. Eqn. 131 (1996) 304-335. | MR | Zbl
[3] and , Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5 (2005) 333-361. | MR | Zbl
[4] and , Dual formulations in critical state problems. Interfaces Free Bound. 8 (2006) 347-368. | MR | Zbl
[5] and , A mixed formulation of the Monge-Kantorovich equations. ESAIM: M2AN 41 (2007) 1041-1060. | Numdam | MR | Zbl
[6] and , A quasi-variational inequality problem in superconductivity. M3AS 20 (2010) 679-706. | MR | Zbl
[7] and , On a dual formulation for the growing sandpile problem. Euro. J. Appl. Math. 20 (2008) 169-185. | MR | Zbl
[8] and , On the collapsing sandpile problem. Commun. Pure Appl. Anal. 10 (2011) 625-638. | MR | Zbl
[9] and , Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). | MR | Zbl
[10] , and , Fast/slow diffusion and collapsing sandpiles. J. Differ. Eqs. 137 (1997) 166-209. | MR | Zbl
[11] , A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121-132. | MR | Zbl
[12] , Real Analysis: Modern Techniques and their Applications, 2nd Edition. Wiley-Interscience, New York (1984). | MR | Zbl
[13] and , Elliptic Partial Differential Equations of Second Order, 2nd Edition. Springer, Berlin (1983). | MR | Zbl
[14] , Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York (1984). | Zbl
[15] , A quasivariational inequality in the problem of filling a shape. U.S.S.R. Comput. Math. Phys. 26 (1986) 74-79. | MR
[16] , A variational model of bulk solids mechanics and free-surface segregation. Chem. Eng. Sci. 48 (1993) 3647-3656.
[17] , Sandpiles and river networks: extended systems with nonlocal interactions. Phys. Rev. E 49 (1994) 1161-1167. | MR
[18] , Variational model for sandpile growth. Eur. J. Appl. Math. 7 (1996) 225-235. | MR | Zbl
[19] and , Quasivariational solutions for first order quasilinear equations with gradient constraint. Arch. Ration. Mech. Anal. 205 (2012) 493-514. | MR | Zbl
[20] , Compact sets in the space Lp(0,T;B). Annal. Math. Pura. Appl. 146 (1987) 65-96. | MR | Zbl
[21] , On the existence of the pressure for solutions of the variational Navier-Stokes equations. J. Math. Fluid Mech. 1 (1999) 225-234. | MR | Zbl
[22] , Mathematical Methods in Plasticity. Gauthier-Villars, Paris (1985).
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