Skip to main content Accessibility help

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

  • John W. Barrett (a1) and Leonid Prigozhin (a2)


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.



Hide All
[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, Amsterdam (2003).
[2] Aronson, G., Evans, L.C. and Wu, Y., Fast/slow diffusion and growing sandpiles. J. Differ. Eqn. 131 (1996) 304335.
[3] Bahriawati, C. and Carstensen, C., Three Matlab implementations of the lowest-order Raviart–Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5 (2005) 333361.
[4] Barrett, J.W. and Prigozhin, L., Dual formulations in critical state problems. Interfaces Free Bound. 8 (2006) 347368.
[5] Barrett, J.W. and Prigozhin, L., A mixed formulation of the Monge-Kantorovich equations. ESAIM: M2AN 41 (2007) 10411060.
[6] Barrett, J.W. and Prigozhin, L., A quasi-variational inequality problem in superconductivity. M3AS 20 (2010) 679706.
[7] Dumont, S. and Igbida, N., On a dual formulation for the growing sandpile problem. Euro. J. Appl. Math. 20 (2008) 169185.
[8] Dumont, S. and Igbida, N., On the collapsing sandpile problem. Commun. Pure Appl. Anal. 10 (2011) 625638.
[9] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).
[10] Evans, L.C., Feldman, M. and Gariepy, R.F., Fast/slow diffusion and collapsing sandpiles. J. Differ. Eqs. 137 (1997) 166209.
[11] Farhloul, M., A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121132.
[12] G.B. Folland, Real Analysis: Modern Techniques and their Applications, 2nd Edition. Wiley-Interscience, New York (1984).
[13] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition. Springer, Berlin (1983).
[14] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York (1984).
[15] Prigozhin, L., A quasivariational inequality in the problem of filling a shape. U.S.S.R. Comput. Math. Phys. 26 (1986) 7479.
[16] Prigozhin, L., A variational model of bulk solids mechanics and free-surface segregation. Chem. Eng. Sci. 48 (1993) 36473656.
[17] Prigozhin, L., Sandpiles and river networks: extended systems with nonlocal interactions. Phys. Rev. E 49 (1994) 11611167.
[18] Prigozhin, L., Variational model for sandpile growth. Eur. J. Appl. Math. 7 (1996) 225235.
[19] Rodrigues, J.F. and Santos, L., Quasivariational solutions for first order quasilinear equations with gradient constraint. Arch. Ration. Mech. Anal. 205 (2012) 493514.
[20] Simon, J., Compact sets in the space L p(0,T;B). Annal. Math. Pura. Appl. 146 (1987) 6596.
[21] Simon, J., On the existence of the pressure for solutions of the variational Navier-Stokes equations. J. Math. Fluid Mech. 1 (1999) 225234.
[22] R. Temam, Mathematical Methods in Plasticity. Gauthier-Villars, Paris (1985).


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

  • John W. Barrett (a1) and Leonid Prigozhin (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed