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 , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1133-1165. doi : 10.1051/m2an/2012062. http://archive.numdam.org/articles/10.1051/m2an/2012062/
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