American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the asymptotic behavior of elliptic, anisotropic singular perturbations problems
  • CPAA Home
  • This Issue
  • Next Article
    A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence
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 and Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161
[1]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial and Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115
[2]

Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078
[3]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 903-920. doi: 10.3934/dcdsb.2021073
[4]

Luis Bayón, Jose Maria Grau, Maria del Mar Ruiz, Pedro Maria Suárez. A hydrothermal problem with non-smooth Lagrangian. Journal of Industrial and Management Optimization, 2014, 10 (3) : 761-776. doi: 10.3934/jimo.2014.10.761
[5]

Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure and Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237
[6]

Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307
[7]

Fatemeh Bazikar, Saeed Ketabchi, Hossein Moosaei. Smooth augmented Lagrangian method for twin bounded support vector machine. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021027
[8]

Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[9]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[10]

Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems and Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237
[11]

Yuan Shen, Wenxing Zhang, Bingsheng He. Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints. Journal of Industrial and Management Optimization, 2014, 10 (3) : 743-759. doi: 10.3934/jimo.2014.10.743
[12]

Xihong Yan. An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game. Journal of Industrial and Management Optimization, 2016, 12 (3) : 879-890. doi: 10.3934/jimo.2016.12.879
[13]

Li Jin, Hongying Huang. Differential equation method based on approximate augmented Lagrangian for nonlinear programming. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2267-2281. doi: 10.3934/jimo.2019053
[14]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control and Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011
[15]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control and Related Fields, 2021, 11 (3) : 521-554. doi: 10.3934/mcrf.2020052
[16]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[17]

Paul Glendinning. Non-smooth pitchfork bifurcations. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 457-464. doi: 10.3934/dcdsb.2004.4.457
[18]

Laetitia Paoli. A proximal-like algorithm for vibro-impact problems with a non-smooth set of constraints. Conference Publications, 2011, 2011 (Special) : 1186-1195. doi: 10.3934/proc.2011.2011.1186
[19]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293
[20]

Petteri Harjulehto, Peter Hästö, Juha Tiirola. Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models. Inverse Problems and Imaging, 2015, 9 (3) : 835-851. doi: 10.3934/ipi.2015.9.835

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (135)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy