Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T14:51:09.335Z Has data issue: false hasContentIssue false
  • English
  • Français

A quasi-variational inequality problem arising in the modelingof growing sandpiles

Published online by Cambridge University Press:  17 June 2013

John W. Barrett  and
Leonid Prigozhin
John W. Barrett
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.. j.barrett@imperial.ac.uk
Leonid Prigozhin
Affiliation:
Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 84990, Israel.
Get access

Abstract

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

Keywords

Quasi-variational inequalitiescritical-state problemsprimal and mixed formulationsfinite elementsexistenceconvergence analysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 4: Direct and inverse modeling of the cardiovascular and respiratory systems , July 2013 , pp. 1133 - 1165
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, Amsterdam (2003).
Aronson, G., Evans, L.C. and Wu, Y., Fast/slow diffusion and growing sandpiles. J. Differ. Eqn. 131 (1996) 304–335. Google Scholar
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) 333–361. Google Scholar
Barrett, J.W. and Prigozhin, L., Dual formulations in critical state problems. Interfaces Free Bound. 8 (2006) 347–368. Google Scholar
Barrett, J.W. and Prigozhin, L., A mixed formulation of the Monge-Kantorovich equations. ESAIM: M2AN 41 (2007) 1041–1060. Google Scholar
Barrett, J.W. and Prigozhin, L., A quasi-variational inequality problem in superconductivity. M3AS 20 (2010) 679–706. Google Scholar
Dumont, S. and Igbida, N., On a dual formulation for the growing sandpile problem. Euro. J. Appl. Math. 20 (2008) 169–185. Google Scholar
Dumont, S. and Igbida, N., On the collapsing sandpile problem. Commun. Pure Appl. Anal. 10 (2011) 625–638. Google Scholar
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).
Evans, L.C., Feldman, M. and Gariepy, R.F., Fast/slow diffusion and collapsing sandpiles. J. Differ. Eqs. 137 (1997) 166–209. Google Scholar
Farhloul, M., A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121–132. Google Scholar
G.B. Folland, Real Analysis: Modern Techniques and their Applications, 2nd Edition. Wiley-Interscience, New York (1984).
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition. Springer, Berlin (1983).
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York (1984).
Prigozhin, L., A quasivariational inequality in the problem of filling a shape. U.S.S.R. Comput. Math. Phys. 26 (1986) 74–79. Google Scholar
Prigozhin, L., A variational model of bulk solids mechanics and free-surface segregation. Chem. Eng. Sci. 48 (1993) 3647–3656. Google Scholar
Prigozhin, L., Sandpiles and river networks: extended systems with nonlocal interactions. Phys. Rev. E 49 (1994) 1161–1167. Google ScholarPubMed
Prigozhin, L., Variational model for sandpile growth. Eur. J. Appl. Math. 7 (1996) 225–235. Google Scholar
Rodrigues, J.F. and Santos, L., Quasivariational solutions for first order quasilinear equations with gradient constraint. Arch. Ration. Mech. Anal. 205 (2012) 493–514. Google Scholar
Simon, J., Compact sets in the space L p(0,T;B). Annal. Math. Pura. Appl. 146 (1987) 65–96. Google Scholar
Simon, J., On the existence of the pressure for solutions of the variational Navier-Stokes equations. J. Math. Fluid Mech. 1 (1999) 225–234. Google Scholar
R. Temam, Mathematical Methods in Plasticity. Gauthier-Villars, Paris (1985).