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January  2009, 8(1): 161-178. doi: 10.3934/cpaa.2009.8.161

A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics

Alexandre Caboussat 1, and  Roland Glowinski 1,

1. 

University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, Texas 77204 - 3008, United States

Received  May 2008 Revised  August 2008 Published  October 2008

An operator-splitting algorithm is presented for the solution of a partial differential equation arising in the modeling of deposition processes in sand mechanics. Sand piles evolution is modeled by an advection-diffusion equation, with a non-smooth diffusion operator that contains a point-wise constraint on the gradient of the solution. Piecewise linear finite elements are used for the discretization in space. The advection operator is treated with a stabilized SUPG finite element method. An augmented Lagrangian method is proposed for the discretization of the fast/slow non-smooth diffusion operator. A penalization approach, together with a Newton method, is used for the treatment of inequality constraints. Numerical results are presented for the simulation of sand piles on flat and non-flat surfaces, and for extensions to water flows.
Keywords: finite element approximation, Operator splitting, non-smooth diffusion, augmented Lagrangian method, point-wise constraints, transport equation.
Mathematics Subject Classification: Primary: 65N30, 65M60, 65K10, 49S05; Secondary: 74S05.
Citation: Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161
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