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

John W. Barrett, Leonid Prigozhin

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


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.

Original languageEnglish
Pages (from-to)1133-1165
Number of pages33
JournalESAIM: Mathematical Modelling and Numerical Analysis
Issue number4
StatePublished - 1 Jul 2013


  • Convergence analysis
  • Critical-state problems
  • Existence
  • Finite elements
  • Primal and mixed formulations
  • Quasi-variational inequalities

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Modeling and Simulation
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'A quasi-variational inequality problem arising in the modeling of growing sandpiles'. Together they form a unique fingerprint.

Cite this