2012/03/07 by John W. Barrett, Barrett, John W., Leonid Prigozhin +1 · 1 citation
Earth and Planetary Sciences · Engineering · Mathematics · #Geological formations and processes #Hydraulic Fracturing and Reservoir Analysis #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1203.1611
Existence of a solution to the quasi-variational inequality problem arising\nin a model for sand surface evolution has been an open problem for a long time.\nAnother long-standing open problem concerns determining the dual variable, the\nflux of sand pouring down the evolving sand surface, which is also of practical\ninterest in a variety of applications of this model. Previously, these problems\nwere solved for the special case in which the inequality is simply variational.\nHere, we introduce a regularized mixed formulation involving both the primal\n(sand surface) and dual (sand flux) variables. We derive, analyse and compare\ntwo methods for the approximation, and numerical solution, of this mixed\nproblem. We prove subsequence convergence of both approximations, as the mesh\ndiscretization parameters tend to zero; and hence prove existence of a solution\nto this mixed model and the associated regularized quasi-variational inequality\nproblem. One of these numerical approximations, in which the flux is\napproximated by the divergence-conforming lowest order Raviart-Thomas element,\nleads to an efficient algorithm to compute not only the evolving pile surface,\nbut also the flux of pouring sand. Results of our numerical experiments confirm\nthe validity of the regularization employed.\n