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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