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A quasi-variational inequality problem arising in the modeling of 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.
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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.

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

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