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.