  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.
